Corrigendum to “Finite Type Invariants of w-Knotted Objects II”

By Dror Bar-Natan and Zsuzsanna Dancso

Feb. 29, 2024

(a revised version of the paper is attached below this corrigendum)

In Section 4 of [BND] we introduce and study w-tangled foams. These are defined combinatorially, as a finitely generated circuit algebra with certain Reidemeister relations. For motivation, we present a local topological interpretation of w-tangled foams as tangled tubes in ℝ^{4}. Unfortunately, some errors have occurred in this interpretation, stemming from our lack of care around “2D orientations”, as below. We wish to thank Yusuke Kuno and Haruko Miyazawa for noting these issues.

e  e e  e R4 R4 A_{e}  A_{e}

• If foams are to have 2D orientations, they cannot be glued to each other unless these orientations match. This breaks the circuit algebra structure. This can be corrected by switching to coloured circuit algebras, in which strands and strand-ends are coloured by their 2D orientations, and gluings are allowed only if colours match. However, we choose a different resolution explained below.

• We missed that the “A_{e}” operations of flipping only the 1D but not the 2D orientations of a strand interact unpleasantly with R4 moves, as shown on the right. This can be corrected by giving more care to the colours (2D orientations) of the edges next to a vertex and making the precise meaning of the R4 move depend on these colours. Since this is notationally tedious, we choose a different resolution below.

Although the resolution is a small change to the content of the paper, it necessitates some notation and language changes, and for the convenience of our readers we have incorporated these into the body of the paper as a revised version, available at [BNDv2].

wen

The easier solution to both of the problems above is to forgo of 2D orientations and of the operations A_{e} altogether, and to replace them with more serious attention to the wens, which are cut open Klein-bottles ([BND, Section 4.5], and [BNDv2, Section 4.1]). Indeed, conjugating a strand by a wen and reversing it (strand reversal is the operation S_{e}, unchanged from the original paper) has the same effect on crossings as the operation A_{e} was meant to have.

The main result of the paper is identifying homomorphic expansions for w-tangled foams with solutions to the Kashiwara-Vergne (KV) equations. A crucial equation relating the values of such an expansion, the unitarity equation V · A_{1}A_{2}(V) = 1 [BND, Equation (12)] is replaced by an equivalent equation, V^{∗}V = 1 [BNDv2, Equation (U), page 41] in which the “global adjoint” V^{∗} is obtained from V by multiplying it by a wen on all three sides, and reversing all strand orientations.

In [BND] we claimed that the value of the wen under an expansion must be 1, and that expansions for w-tangled foams are in bijection with Kashiwara-Vergne solutions with even Duflo function. In fact, the value of the wen could be non-trivial [BNDv2, Lemma 4.9], but the statement regarding KV solutions remains true for homomorphic expansions where the value of the wen is set to be 1.

Theorem 4.9 of [BND] stated that for “orientable w-tangled foams”, i.e. where wens are not included, homomorphic expansions are in one to one correspondence with KV solutions in general. This theorem remains true: in [BNDv2, Section 4.6] we present a more careful definition of orientable w-tangled foams as a sub-circuit algebra. Edge-wise adjoint operations replaced by the well-defined global adjoint (global orientation reversal and composition with wens at every tangle end) as above, noting that these wens all cancel. Homomorphic expansions for these foams are indeed in one to one correspondence with KV solutions.

We note that the results of papers [BDS, DHR] which build on Section 4 of [BND] are unaffected by this Corrigendum: in these papers one can simply mechanically replace A_{1}A_{2}(V) with V^{∗}.

Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.

References

[BND] D. Bar-Natan and Z. Dancso, Finite Type Invariants of W-Knotted Objects II: Tangles and the Kashiwara-Vergne Problem, Math. Ann. 367 (2017) 1517–1586, arXiv:1405.1955v3.

[BNDv2] D. Bar-Natan and Z. Dancso, Finite Type Invariants of W-Knotted Objects II: Tangles and the Kashiwara-Vergne Problem, http://drorbn.net/WKO2 and arXiv:1405.1955.

[BDS] D. Bar-Natan, Z. Dancso, N. Scherich, Ribbon 2-knots, 1+1 = 2, and Duflo’s Theorem for Arbitrary Lie Algebras Alg. Geom. Topol. 20(7) 3733–3760. arXiv:1811.08558

[DHR] Z. Dancso, I. Halacheva, M. Robertson, A Topological Characterisation of the Kashiwara–Vergne Groups Trans. Amer. Math. Soc. 376(5), 3265–3317. arXiv:2106.02373

FINITE TYPE INVARIANTS OF W-KNOTTED OBJECTS II (V2): TANGLES, FOAMS AND THE KASHIWARA-VERGNE PROBLEM

DROR BAR-NATAN AND ZSUZSANNA DANCSO

Abstract. This is the second in a series of papers dedicated to studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.). These are classes of knotted objects that are \underline{w}ider but \underline{w}eaker than their “\underline{u}sual” counterparts. To get (say) w-knots from usual knots (or u-knots), one has to allow non-planar “virtual” knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation beyond the ordinary collection of Reidemeister moves, called the “overcrossings commute” relation, making wknotted objects a bit weaker once again. Satoh [Sa] studied several classes of w-knotted objects (under the name “\underline{w}eakly-virtual”) and has shown them to be closely related to certain classes of knotted surfaces in ℝ^{4}.

In this article we study finite type invariants of w-tangles and w-trivalent graphs (also referred to as w-tangled foams). Much as the spaces A of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces A^{w} of “arrow diagrams” for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic universal finite type invariant of w-foams is essentially the same as a solution of the Kashiwara-Vergne [KV] conjecture and much of the Alekseev-Torossian [AT] work on Drinfel’d associators and Kashiwara-Vergne can be re-interpreted as a study of w-foams.

Contents

Conflict of interest statement 2 References 2 1. Introduction 2 1.1. Motivation and hopes 2 1.2. A brief overview and large-scale explanation 3 1.3. Corrigendum 5 1.4. Acknowledgement 5 2. Algebraic Structures, Expansions, and Circuit Algebras 5 2.1. Algebraic Structures 6 2.2. Associated Graded Structures 7 2.3. Expansions and Homomorphic Expansions 8 2.4. Circuit Algebras 9

Date: first edition May 5, 2014, this edition Feb. 29, 2024. Published in Mathematische Annalen 367

(2017) 1517–1586. See http://www.math.toronto.edu/drorbn/LOP.html#WKO2; the arXiv:1405.1955 edi-

tion may be older.

2010 Mathematics Subject Classification. 57M25. Key words and phrases. virtual knots, w-braids, w-knots, w-tangles, knotted graphs, finite type invari-

ants, Alexander polynomial, Kashiwara-Vergne, associators, free Lie algebras.

This work was partially supported by NSERC grant RGPIN 262178. This paper is part 2 of a 4-part series whose first two parts originally appeared as a combined preprint, [WKO0].

3. w-Tangles 12 3.1. v-Tangles and w-Tangles 12 3.2. A^{w}(↑_{n}) and the Alekseev-Torossian Spaces 18 3.3. The Relationship with u-Tangles 24 3.4. The local topology of w-tangles 26 3.5. Good properties and uniqueness of the homomorphic expansion 28 4. w-Tangled Foams 30 4.1. The Circuit Algebra of w-Tangled Foams 30 4.2. The local topology of w-tangled foams 31 4.3. The Associated Graded Structure 35 4.4. The homomorphic expansion 37 4.5. The equivalence with the Alekseev-Torossian equations 41 4.6. Orientable w-tangled foams 43 4.7. Interlude: u-Knotted Trivalent Graphs 45 4.8. The relationship between sKTG and wTF^{o} 50 5. Odds and Ends 55 5.1. Motivation for circuit algebras: electronic circuits 55 5.2. Proof of Proposition 4.25 55 6. Glossary of notation 58 References 59

1. Introduction

This is the second in a series of papers on w-knotted objects. In the first paper [WKO1], we took a classical approach to studying finite type invariants of w-braids and w-knots and proved that the universal finite type invariant for w-knots is essentially the Alexander polynomial. In this paper we will study finite type invariants of w-tangles and w-tangled foams from a more algebraic point of view, and prove that “homomorphic” universal finite type invariants of w-tangled foams are in one-to-one correspondence with solutions to the (Alekseev-Torossian version of) the Kashiwara-Vergne problem in Lie theory. Mathematically, this paper does not depend on the results of [WKO1] in any significant way, and the reader familiar with the theory of finite type invariants will have no difficulty reading this paper without having read [WKO1]. However, since this paper starts with an abstract rephrasing of the well-known finite type story in terms of general algebraic structures, readers who need an introduction to finite type invariants may find it more pleasant to read [WKO1] first (especially Sections 1, 2 and 3.1–3.5).

1.1. Motivation and hopes. This article and its siblings [WKO1] and [WKO3] are efforts towards a larger goal. Namely, we believe many of the difficult algebraic equations in mathematics, especially those that are written in graded spaces, more especially those that are related in one way or another to quantum groups [Dr1], and to the work of Etingof and Kazhdan [EK], can be understood, and indeed would appear more natural, in terms of finite type invariants of various topological objects.

This work was inspired by Alekseev and Torossian’s results [AT] on Drinfel’d associators and the Kashiwara-Vergne conjecture, both of which fall into the aforementioned class of “difficult equations in graded spaces”. The Kashiwara-Vergne conjecture — proposed in 1978 [KV] and proven in 2006 by Alekseev and Meinrenken [AM] — has strong implications in Lie theory and harmonic analysis, and is a cousin of the Duflo isomorphism, which was shown to be knot-theoretic in [BLT]. We also know that Drinfel’d’s theory of associators [Dr2] can be interpreted as a theory of well-behaved universal finite type invariants of parenthesized tangles^{1} [LM, BN2], or of knotted trivalent graphs [Da].

In Section 4 we will re-interpret the Kashiwara-Vergne conjecture as the problem of finding a “homomorphic” universal finite type invariant of a class of w-knotted trivalent graphs (more accurately named w-tangled foams). This result fits into a bigger picture incorporating usual, virtual and w-knotted objects and their theories of finite type invariants, connected by the inclusion map from usual to virtual, and the projection from virtual to w-knotted objects. In a sense that will be made precise in Section 2, usual and w-knotted objects with this mapping form a unified algebraic structure, and the relationship between Drinfel’d associators and the Kashiwara-Vergne conjecture is explained as a theory of finite type invariants for this larger structure. This will be the topic of Section 4.7.

We are optimistic that this paper is a step towards re-interpreting the work of Etingof and Kazhdan [EK] on quantization of Lie bi-algebras as a construction of a well-behaved universal finite type invariant of virtual knots [Ka, Kup] or of a similar class of virtually knotted objects. However, w-knotted objects are quite interesting in their own right, both topologically and algebraically: they are related to combinatorial group theory, to groups of movies of flying rings in ℝ^{3}, and more generally, to certain classes of knotted surfaces in ℝ^{4}. The references include [BH, FRR, Gol, Mc, Sa].

In [WKO1] we studied the universal finite type invariants of w-braids and w-knots, the latter of which turns out to be essentially the Alexander polynomial. A more thorough introduction about our “hopes and dreams” and the u-v-w big picture can also be found in [WKO1].

1.2. A brief overview and large-scale explanation. We are going to start by developing the algebraic ingredients of the paper in Section 2. The general notion of an algebraic structure lets us treat spaces of a topological or diagrammatic nature in a unified algebraic manner. All of braids, w-braids, w-knots, w-tangles, etc., and their associated chord- or arrow-diagrammatic counterparts form algebraic structures, and so do any number of these spaces combined, with maps between them.

We then introduce associated graded structures with respect to a specific filtration, the machine which in our case takes an algebraic structure of “topological nature” (say, braids with n strands) and produces the corresponding diagrammatic space (for braids, horizontal chord diagrams on n vertical strands). This is done by taking the associated graded space with respect to a given filtration, namely the powers of the augmentation ideal in the algebraic structure.

An expansion, sometimes called a universal finite type invariant, is a map from an algebraic structure (in this case one of topological nature) to its associated graded (a structure of combinatorial/diagrammatic nature), with a certain universality property. A homomorphic expansion is one that is in addition “well behaved” with respect to the operations of the algebraic structure (such as composition and strand doubling for braids, for example).

The three main results of the paper are as follows:

1“q-tangles” in [LM], “non-associative tangles” in [BN2].

(1) As mentioned before, our goal is to provide a topological framework for the Kashiwara-Vergne (KV) problem. One of our main results is Theorem 4.23, in which we establish a bijection between certain homomorphic expansions of w-tangled foams (introduced in Section 4) and solutions of the Kashiwara-Vergne equations. More precisely, “certain” homomorphic expansions means ones that are group-like (a commonly used condition), and subject to another minor technical condition. Section 3 leads up to this result by studying the simpler case of w-tangles and identifying building blocks of its associated graded structure as the spaces which appear in the [AT] formulation of the KV equations.

(2) In Theorem 4.8 we study an unoriented version of w-tangled foams, and prove that homomorphic expansions for this space (group-like and subject to the some minor conditions) are in one-to-one correspondence with solutions to the KV problem with even Duflo function. This sets the stage for perhaps the most interesting result of the paper:

(3) Section 4.8 marries the theory above with the theory of ordinary (not w-) knotted trivalent graphs (KTGs). For technical reasons explained in Section 4, we work with a signed version of KTGs (sKTG). Roughly speaking, homomorphic expansions for sKTGs are determined by a Drinfel’d associator. Furthermore, sKTGs map naturally into w-tangled foams. In Theorem 4.27 we prove that any homomorphic expansion of sKTGs coming from a horizontal chord associator has a compatible homomorphic expansion of w-tangled foams, and furthermore, these expansions are in one-to-one correspondence with symmetric solutions of the KV problem. This gives a topological explanation for the relationship between Drinfel’d associators and the KV conjecture.

We note that in [WKO3] we’ll further capitalize on these insights to provide a topological proof and interpretation for Alekseev, Enriquez and Torossian’s explicit solutions for the KV conjecture in terms of associators [AET].

Several of the structures of a topological nature in this paper (w-tangles and w-foams) are introduced as Reidemeister theories. That is, the spaces are built from pictorial generators (such as crossings) which can be connected arbitrarily, and the resulting pictures are then factored out by certain relations (“Reidemeister moves”). Technically speaking, this is done using the framework of circuit algebras (similar to planar algebras but without the planarity requirement) which are introduced in Section 2.

One of the fundamental theorems of classical knot theory is Reidemeister’s theorem, which states that isotopy classes of knots are in bijection with knot diagrams modulo Reidemeister moves. In our case, w-knotted objects have a Reidemeister description and a topological interpretation in terms of ribbon knotted tubes in ℝ^{4}. However, the analogue of the Reidemeister theorem, i.e. the statement that these two interpretations coincide, is only known for w-braids [Mc, D, BH].

For w-tangles and w-foams (and w-knots as well) there is a map δ from the Reidemeister presentation to the appropriate class of ribbon 2-knotted objects in ℝ^{4}. In our case this means that all the generators have a local topological interpretation and the relations represent isotopies. The map δ is certainly a surjection, but it is only conjectured to be injective (in other words, it is possible that some relations are missing).

The main difficulty in proving the injectivity of δ lies in the management of the ribbon structure. A ribbon 2-knot is a knotted sphere or long tube in ℝ^{4} which admits a filling with only certain types of singularities. While there are Reidemeister theorems for general 2-knots in ℝ^{4} [CS], the techniques don’t translate well to ribbon 2-knots, mainly because it is not well understood how different ribbon structures (fillings) of the same ribbon 2-knot can be obtained from each other through Reidemeister type moves. The completion of such a theorem would be of great interest. We suspect that even if δ is not injective, the present set of generators and relations describes a set of ribbon-knotted tubes in ℝ^{4} with possibly some extra combinatorial information, similarly to how, say, dropping the R1 relation in classical knot theory results in a Reidemeister theory for framed knots with rotation numbers.

The paper is organized as follows: we start with a discussion of general algebraic structures, associated graded structures, expansions (universal finite type invariants) and “circuit algebras” in Section 2. In Section 3 we study w-tangles and identify some of the spaces [AT] where the KV conjecture “lives” as the spaces of “arrow diagrams” (the w-analogue of chord diagrams) for certain w-tangles. In Section 4 we study w-tangled foams and we prove the main theorems discussed above. For more detailed information consult the “Section Summary” paragraphs at the beginning of each of the sections. A glossary of notation is on page 58.

1.3. Corrigendum. Yusuke Kuno and Haruko Miyazawa pointed out an error in the topological interpretation of w-tangled foams. Following several in-depth conversations with them, we made improvements to Section 4, and submitted a Corrigendum [WKO2C]. In short, the “adjoint” A_{e} operations of the published version of this paper were not welldefined with respect to the Reidemeister 4 relation. The simplest fix is to omit surface (two-dimensional) orientations and the adjoint operations entirely. The adjoint operations were used to relate the two “vertex” generators of w-foams, however, this can be achieved just as well by composing one of the vertices by “wens” (cut open Klein bottles). In this version of the paper we incorporated these changes into Section 4 for easier reading, and made several other stylistic improvements to the exposition.

1.4. Acknowledgement. We thank Yusuke Kuno and Haruko Miyazawa for their significant contribution which helped us improve Section 4 post-publication. We wish to thank Anton Alekseev, Jana Archibald, Scott Carter, Karene Chu, Iva Halacheva, Joel Kamnitzer, Lou Kauffman, Peter Lee, Louis Leung, Jean-Baptiste Meilhan, Dylan Thurston, Lucy Zhang and the anonymous referees for comments and suggestions.

2. Algebraic Structures, Expansions, and Circuit Algebras

Section Summary. In this section we introduce the associated graded structure of an “arbitrary algebraic structure” with respect to powers of its augmentation ideal (Sections 2.1 and 2.2) and introduce the notions of “expansions” and “homomorphic expansions” (2.3). Everything is so general that practically anything is an example, yet our main goal is to set the language for the examples of w-tangles and w-tangled foams, which appear later in this paper. Both of these examples are types of “circuit algebras”, and hence we end this section with a general discussion of circuit algebras (Sec. 2.4).

2.1. Algebraic Structures. An “algebraic structure” O is some collection (O_{α}) of sets of objects of different kinds, where the subscript α denotes the “kind” of the objects in O_{α}, along with some collection of “operations” ψ_{β}, where each ψ_{β} is an arbitrary map with domain some product O_{α_{1}} × ⋯ × O_{α_{k}} of sets of objects, and range a single set O_{α_{0}} (so operations may be unary or binary or multinary, but they always return a value of some fixed kind). We also allow some named “constants” within some O_{α}’s (or equivalently, allow some 0-nary operations).^{2} The operations may or may not be subject to axioms — an “axiom” is an identity asserting that some composition of operations is equal to some other composition of operations.

Figure 1. An algebraic structure O with 4 kinds of objects and one binary, 3 unary and two 0-nary operations (the constants 1 and

σ). { objects of kind 3 } = O =  O_{3}  O_{4} O_{1}^{•1}  O_{2}^{•σ} ψ_{1}  ψ_{3}  ψ_{4} ψ_{2}

Figure 1 illustrates the general notion of an algebraic structure. Here are a few specific examples:

• We will use ⟨b⟩, the free group on one generator b, as a running example throughout this chapter (of course ⟨b⟩ is isomorphic to ℤ). This is an algebraic structure with one kind of objects, a binary operation “multiplication”, a unary operation “inverse”, one constant “the identity”, and the expected axioms.

• Groups in general: one kind of objects, one binary “multiplication”, one unary “inverse”, one constant “the identity”, and some axioms.

• Group homomorphisms: Two kinds of objects, one for each group. 7 operations — 3 for each of the two groups and the homomorphism itself, going between the two groups. Many axioms.

• A group acting on a set, a group extension, a split group extension and many other examples from group theory.

• A quandle is a set with an operation ↑, satisfying (x ↑ y) ↑ z = (x ↑ y) ↑ (y ↑ z) and some further minor axioms. This is an algebraic structure with one kind of objects and one operation. See [WKO0] for an analysis of quandles from the perspective of this paper.

• Planar algebras as in [Jon] and circuit algebras as in Section 2.4.

• The algebra of knotted trivalent graphs as in [BN4, Da].

• Let ς: B → S be an arbitrary homomorphism of groups (though our notation suggests what we have in mind — B may well be braids, and S may well be permutations). We can consider an algebraic structure O whose kinds are the elements of S, for which the objects of kind s ∈ S are the elements of O_{s} := ς^{−1}(s), and with the product in B defining operations O_{s_{1}} × O_{s_{2}} → O_{s_{1}s_{2}}.

2Alternatively define “algebraic structures” using the theory of “multicategories” [Lei]. Using this language, an algebraic structure is simply a functor from some “structure” multicategory C into the multicategory Set (or into Vect, if all O_{i} are vector spaces and all operations are multi-linear). A “morphism”

between two algebraic structures over the same multicategory C is a natural transformation between the two functors representing those structures.

• W-tangles and w-foams, studied in the following two sections of this paper.

• Clearly, many more examples appear throughout mathematics.

2.2. Associated Graded Structures. Any algebraic structure O has an “especially natural” associated graded structure: that is, we take the associated structure with respect to a specific and natural filtration. This will be a repeating construction throughout the rest of this paper series.

First extend O to allow formal linear combinations of objects of the same kind (extending the operations in a linear or multi-linear manner), then let I, the “augmentation ideal”, be the sub-structure made out of all such combinations in which the sum of coefficients is 0, then let I^{m} be the set of all outputs of algebraic expressions (that is, arbitrary compositions of the operations in O) that have at least m inputs in I (and possibly, further inputs in O), and finally, set

grad O := ⨁_{m≥0} I^{m}/I^{m+1}.  (1)

Clearly, with the operations inherited from O, the associated graded grad O is again algebraic structure with the same multi-graph of spaces and operations, but with new objects and with new operations that may or may not satisfy the axioms satisfied by the operations of O. The main new feature in grad O is that it is a “graded” structure; we denote the degree m piece I^{m}/I^{m+1} of grad O by grad_{m}O.

We believe that many of the most interesting graded structures that appear in mathematics are the result of this construction (i.e., as associated graded structures with respect to powers of the augmentation ideal), and that many of the interesting graded equations that appear in mathematics arise when one tries to find “expansions”, or “universal finite type invariants”, which are also morphisms^{3} Z: O → grad O (see Section 2.3) or when one studies “automorphisms” of such expansions^{4}. Indeed, the paper you are reading now is really the study of the associated graded structures of various algebraic structures associated with wknotted objects. We would like to believe that much of the theory of quantum groups (at “generic” ℏ) will eventually be shown to be a study of the associatead graded structures of various algebraic structures associated with v-knotted objects.

Example 2.1. We compute the associated graded structuture of the running example ⟨b⟩. Allowing formal ℚ-linear combinations of elements we get ℚ⟨b⟩ = ℚ[b, b^{−1}]. The augmentation ideal I is generated by differences (b^{n} − 1) as a vector space (where 1 = b^{0}), and generated by (b − 1) as an ideal.

We claim that grad ⟨b⟩ ≅ ℚ[[c]], the algebra of power series in one variable. To show this, consider the map π : ℚ[[c]] → grad ⟨b⟩ by setting π(c) = [b − 1] (mod I^{2}). It is easy to show explicitly that π is surjective. For example, in degree 1, we need to show that b−1 generates I/I^{2}. indeed, (b^{n} − 1) − n(b − 1) has a double zero at b = 1, and hence f = \frac{(b^{n}−1)−n(b−1)}{(b−1)^{2}} is a polynomial, and b^{n} − 1 = n(b − 1) + f(b − 1)^{2}. So modulo (b − 1)^{2} ∈ I^{2}, b^{n} − 1 = n(b − 1). A similar argument works to show that (b − 1)^{k} generates I^{k}/I^{k+1}.

3Indeed, if O is finitely presented then finding such a morphism Z: O → grad O amounts to finding its values on the generators of O, subject to the relations of O. Thus it is equivalent to solving a system of equations written in some graded spaces.

4The Drinfel’d graded Grothendieck-Teichmuller group GRT is an example of such an automorphism group. See [Dr3, BN3].

Note that ⟨b⟩ can also be thought of as the pure braid group on two strands: b would be a “full twist” and c can be represented as a single “horizontal chord”. In other knot theoretic settings, it is generally relatively easy to find a “candidate associated graded” and a map π, which can be shown to be surjective by explicit means.

To show that π is injective we are going to use the machinery of “expansions” which is the tool we use to accomplish similar tasks in the later sections of this paper.

We end this section with two more examples of computing associated graded structures: the proof of Proposition 2.2 is an exercise; for the proof of Proposition 2.3 see [WKO0].

Proposition 2.2. If G is a group, grad G is a graded associative algebra with unit. Similarly, the associated graded structure of a group homomorphism is a homomorphism of graded associative algebras. □

Proposition 2.3. If Q is a unital quandle, grad_{0}Q is one-dimensional and grad_{>0}Q is a graded right Leibniz algebra^{5} generated by grad_{1}Q.

2.3. Expansions and Homomorphic Expansions. We start with the definition. Given an algebraic structure O let fil O denote the filtered structure of linear combinations of objects in O (respecting kinds), filtered by the powers (I^{m}) of the augmentation ideal I. Recall also that any graded space G = ⨁_{m} G_{m} is automatically filtered, by (⨁_{n≥m} G_{n})_{m=0}^{∞}.

Definition 2.4. An “expansion” Z for O is a map Z: O → grad O that preserves the kinds of objects and whose linear extension (also called Z) to fil O respects the filtration of both sides, and for which (gr Z) : (gr fil O = grad O) → (gr grad O = grad O) is the identity map of grad O; we refer to this as the “universality property”.

In practical terms, this is equivalent to saying that Z is a map O → grad O whose restriction to I^{m} vanishes in degrees less than m (in grad O) and whose degree m piece is the projection I^{m} → I^{m}/I^{m+1}.

We come now to what is perhaps the most crucial definition in this paper.

Definition 2.5. A “homomorphic expansion” is an expansion which also commutes with all the algebraic operations defined on the algebraic structure O.

Why Bother with Homomorphic Expansions? Primarily, for two reasons:

• Often grad O is simpler to work with than O; for one, it is graded and so it allows for finite “degree by degree” computations, whereas often times, such as in many topological examples, anything in O is inherently infinite. Thus it can be beneficial to translate questions about O to questions about grad O. A simplistic example would be, “is some element a ∈ O the square (relative to some fixed operation) of an element b ∈ O?”. Well, if Z is a homomorphic expansion and by a finite computation it can be shown that Z(a) is not a square already in degree 7 in grad O, then we’ve given a conclusive negative answer to the example question. Some less simplistic and more relevant examples appear in [BN4].

• Often grad O is “finitely presented”, meaning that it is generated by some finitely many elements g_{1}, …, g_{k} ∈ O, subject to some relations R_{1} … R_{n} that can be written in terms of g_{1}, …, g_{k} and the operations of O. In this case, finding a homomorphic expansion Z is essentially equivalent to guessing the values of Z on g_{1}, …, g_{k}, in such a manner that these values Z(g_{1}), …, Z(g_{k}) would satisfy the grad O versions of the relations R_{1} … R_{n}. So finding Z amounts to solving equations in graded spaces. It is often the case (as will be demonstrated in this paper; see also [BN2, BN3]) that these equations are very interesting for their own algebraic sake, and that viewing such equations as arising from an attempt to solve a problem about O sheds further light on their meaning.

5A Leibniz algebra is a Lie algebra without anti-commutativity, as defined by Loday in [Lod].

In practice, often the first difficulty in searching for an expansion (or a homomorphic expansion) Z: O → grad O is that its would-be target space grad O is hard to identify. It is typically easy to make a suggestion A for what grad O could be. It is typically easy to come up with a reasonable generating set D_{m} for I^{m} (keep some knot theoretic examples in mind, or ℤ in Example 2.1). It is a bit harder but not exceedingly difficult to discover some relations R satisfied by the elements of the image of D in I^{m}/I^{m+1} (4T, \overrightarrow{4T}, and more in knot theory, there are no relations for ℤ). Thus we set A := D/R; but it is often very hard to be sure that we found everything that ought to go in R; so perhaps our suggestion A is still too big? Finding 4T for example was actually not that easy. Could we have missed some further relations that are hiding in A?

The notion of an A-expansion, defined below, solves two problems at once. Once we find an A-expansion we know that we’ve identified grad O correctly, and we automatically get what we really wanted, a (grad O)-valued expansion.

\xymatrix{ & A \ar[d]_{π} \\ O \ar[ur]^{Z_{A}} \ar[r]_{Z} & grad O \ar[u]_{gr Z_{A}}}

Definition 2.6. A “candidate assoctaed graded structure” for an algebraic structure O is a graded structure A with the same operations as O along with a homomorphic surjective graded map π: A → grad O. An “A-expansion” is a kind and filtration respecting map Z_{A}: O → A for which (gr Z_{A}) ∘ π: A → A is the identity. One can similarly define “homomorphic A-expansions”.

Proposition 2.7. If A is a candidate associated graded of O and Z_{A}: O → A is a homomorphic A-expansion, then π : A → grad O is an isomorphism and Z := π ∘ Z_{A} is a homomorphic expansion. (Often in this case, A is identified with grad O and Z_{A} is identified with Z).

Proof. Note that π is surjective by birth. Since (gr Z_{A}) ∘ π is the identity, π it is also injective and hence it is an isomorphism. The rest is immediate. □

Example 2.8. Back to ⟨b⟩, in Example 2.1 we found a candidate associated graded structure A = ℚ[[c]] and a map π : c ↦ [b − 1]. According to Proposition 2.7, it is enough to find a homomorphic A-expansion, that is, an algebra homomorphism Z_{A} : ℚ⟨b⟩ → ℚ[[c]] such that gr Z_{A} ∘ π is the identity of ℚ[[c]]. It is a straightforward calculation to check that any algebra map defined by Z_{A}(b) = 1 + c + {higher order terms} satisfies this property. If one seeks a “group-like” homomorphic expansion then Z_{A}(b) = e^{c} is the only solution. In either case, exhibiting Z_{A} proves that π is injective and hence A is the associated graded structure of ⟨b⟩.

2.4. Circuit Algebras. “Circuit algebras” are so common and everyday, and they make such a useful language (definitely for the purposes of this paper, but also elsewhere), we find it hard to believe they haven’t made it into the standard mathematical vocabulary^{6}. People familiar with planar algebras [Jon] may note that circuit algebras are just the same as planar algebras, except with the planarity requirement dropped from the “connection diagrams” (and all colourings are dropped as well).

In our context, the main utility of circuit algebras is that they allow for a much simpler presentation of v(irtual)- and w-tangles. There are planar algebra presentations of v- and w-tangles, generated by the usual crossings and the “virtual crossing”, modulo the usual as well as the “virtual” and “mixed” Reidemeister moves. Switching from planar algebras to circuit algebras however renders the extra generators and relations unnecessary: the “virtual crossing” becomes merely a circuit algebra artifact, and the new Reidemeister moves are implied by the circuit algebra structure (see Warning 3.3, Definition 3.4, and Remark 3.5).

The everyday intuition for circuit algebras comes from electronic circuits, whose components can be wired together in many, not necessarily planar, ways, and it is not important to know how these wires are embedded in space. For details and more motivation see Section 5.1. We start formalizing this image by defining “wiring diagrams”, the abstract analogs of printed circuit boards. Let ℕ denote the set of natural numbers including 0, and for n ∈ ℕ let \underline{n} denote some fixed set with n elements, say \{1, 2, …, n\}.

1 2  3  5 4 6 3 1 21 34 1 4 2 4 3 2

Definition 2.9. Let k, n, n_{1}, …, n_{k} ∈ ℕ be natural numbers. A “wiring diagram” D with inputs \underline{n_{1}}, … \underline{n_{k}} and outputs \underline{n} is an unoriented compact 1-manifold whose boundary is \underline{n} ⨿ \underline{n_{1}} ⨿ ⋯ ⨿ \underline{n_{k}}, regarded up to homeomorphism (on the right is an example with k = 3, n = 6, and n_{1} = n_{2} = n_{3} = 4). In strictly combinatorial terms, it is a pairing^{7} of the elements of the set \underline{n} ⨿ \underline{n_{1}} ⨿ ⋯ ⨿ \underline{n_{k}} along with a single further natural number that counts closed circles. If D_{1}; …; D_{m} are wiring diagrams with inputs \underline{n_{11}}, …, \underline{n_{1k_{1}}}; …; \underline{n_{m1}}, …, \underline{n_{mk_{m}}} and outputs \underline{n_{1}}; …; \underline{n_{m}} and D is a wiring diagram with inputs \underline{n_{1}}; …; \underline{n_{m}} and outputs \underline{n}, there is an obvious “composition” D(D_{1}, …, D_{m}) (obtained by gluing the corresponding 1-manifolds, and also describable in completely combinatorial terms) which is a wiring diagram with inputs (\underline{n_{ij}})_{1≤i≤k_{j},1≤j≤m} and outputs \underline{n} (note that closed circles may be created in D(D_{1}, …, D_{m}) even if none existed in D and in D_{1}; …; D_{m}).

A circuit algebra is an algebraic structure (in the sense of Section 2.2) whose operations are parametrized by wiring diagrams. Here’s a formal definition:

Definition 2.10. A circuit algebra consists of the following data:

• For every natural number n ≥ 0 a set (or a ℤ-module) C_{n} “of circuits with n legs”.

• For any wiring diagram D with inputs \underline{n_{1}}, … \underline{n_{k}} and outputs \underline{n}, an operation (denoted by the same letter) D: C_{n_{1}} × ⋯ × C_{n_{k}} → C_{n} (or linear D: C_{n_{1}} ⊗ ⋯ ⊗ C_{n_{k}} → C_{n} if we work with ℤ-modules).

We insist that the obvious “identity” wiring diagrams with \underline{n} inputs and \underline{n} outputs act as the identity of C_{n}, and that the actions of wiring diagrams be compatible in the obvious sense with the composition operation on wiring diagrams.

6Or have they, and we have been looking the wrong way?

7We mean “pairing” in the sense of combinatorics, not in the sense of linear algebra. That is, an involution without fixed point.

A silly but useful example of a circuit algebra is the circuit algebra S of empty circuits, or in our context, of “skeletons”. The circuits with n legs for S are wiring diagrams with n outputs and no inputs; namely, they are 1-manifolds with boundary \underline{n} (so n must be even).

More generally one may pick some collection of “basic components” (analogous to logic gates and junctions for electronic circuits as in Figure 23) and speak of the “free circuit algebra” generated by these components; even more generally we can speak of circuit algebras given in terms of “generators and relations”. (In the case of electronics, our relations may include the likes of De Morgan’s law ¬(p ∨ q) = (¬p) ∧ (¬q) and the laws governing the placement of resistors in parallel or in series.) We feel there is no need to present the details here, yet many examples of circuit algebras given in terms of generators and relations appear in this paper, starting with the next section. We will use the notation C = CA⟨ G | R ⟩ to denote the circuit algebra generated by a collection of elements G subject to some collection R of relations.

People familiar with electric circuits know that connectors sometimes come in “male” and “female” versions, and that you can’t plug a USB cable into a headphone jack. Thus one may define “directed circuit algebras” in which the wiring diagrams are oriented, the circuit sets C_{n} get replaced by C_{p,q} for “circuits with p incoming wires and q outgoing wires” and only orientation preserving connections are ever allowed^{8}. Likewise there is a “coloured” version of everything, in which the wires may be coloured by the elements of some given set X (which may include among its members the elements “USB” and “audio”) and in which connections are allowed only if the colour coding is respected. We will leave the formal definitions of directed and coloured circuit algebras, as well as the definitions of directed and coloured analogues of the skeletons algebra S and generators and relations for directed and coloured algebras, as an exercise.

Note that there is an obvious notion of “a morphism between two circuit algebras” and that circuit algebras (directed or not, coloured or not) form a category. We feel that a precise definition is not needed. A lovely example is the “implementation morphism” of logic circuits in the style of Figure 23 in Section 5 into more basic circuits made of transistors and resistors.

Perhaps the prime mathematical example of a circuit algebra is tensor algebra. If t_{1} is an element (a “circuit”) in some tensor product of vector spaces and their duals, and t_{2} is the same except in a possibly different tensor product of vector spaces and their duals, then once an appropriate pairing D (a “wiring diagram”) of the relevant vector spaces is chosen, t_{1} and t_{2} can be contracted (“wired together”) to make a new tensor D(t_{1}, t_{2}). The pairing D must pair a vector space with its own dual, and so this circuit algebra is coloured by the set of vector spaces involved, and directed, by declaring (say) that some vector spaces are of one gender and their duals are of the other. We have in fact encountered this circuit algebra in [WKO1, Section 3.5].

Let G be a group. A G-graded algebra A is a collection {A_{g}: g ∈ G} of vector spaces, along with products A_{g} ⊗ A_{h} → A_{gh} that induce an overall structure of an algebra on A := ⨁_{g∈G} A_{g}. In a similar vein, we define the notion of an S-graded circuit algebra:

Definition 2.11. An S-graded circuit algebra, or a “circuit algebra with skeletons”, is an algebraic structure C with spaces C_{β}, one for each element β of the circuit algebra of skeletons S, along with composition operations D_{β_{1},...,β_{k}}: C_{β_{1}} × ⋯ × C_{β_{k}} → C_{β}, defined whenever D is a wiring diagram and β = D(β_{1}, …, β_{k}), so that with the obvious induced structure, ∐_{β} C_{β} is a circuit algebra. A similar definition can be made if/when the skeletons are taken to be directed or coloured.

8By convention we label the boundary points of such circuits 1, …, p + q, with the first p labels reserved for the incoming wires and the last q for the outgoing. The inputs of wiring diagrams must be labeled in the opposite way for the numberings to match.

Loosely speaking, a circuit algebra with skeletons is a circuit algebra in which every element T has a well-defined skeleton ς(T) ∈ S. Yet note that as an algebraic structure a circuit algebra with skeletons has more “spaces” than an ordinary circuit algebra, for its spaces are enumerated by skeleta and not merely by integers. The prime examples for circuit algebras with skeletons appear in the next section.

3. w-Tangles

Section Summary. In Sec. 3.1 we introduce v-tangles and w-tangles, the vand w- counterparts of the standard knot-theoretic notion of “tangles”, and briefly discuss their finite type invariants and their associated spaces of “arrow diagrams”, A^{v}(↑_{n}) and A^{w}(↑_{n}). We then construct a homomorphic expansion Z, or a “wellbehaved” universal finite type invariant for w-tangles. The only algebraic tool we need to use is exp(a) := ∑ a^{n}/n! (Sec. 3.1 is in fact a routine extension of parts of [WKO1, Section 3]). In Sec. 3.2 we show that A^{w}(↑_{n}) ≅ U(𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n} ⋉ 𝔱𝔯_{n}), where 𝔞_{n} is an Abelian algebra of rank n and where 𝔱𝔡𝔢𝔯_{n} and 𝔱𝔯_{n}, two of the primary spaces used by Alekseev and Torossian [AT], have simple descriptions in terms of cyclic words and free Lie algebras. We also show that some functionals studied in [AT], div and j, have natural interpretations in our language. In Section 3.3 we discuss a subclass of w-tangles called “special” w-tangles, and relate them by similar means to Alekseev and Torossian’s 𝔰𝔡𝔢𝔯_{n} and to “tree level” ordinary Vassiliev theory. A local topological interpretation is described in Sec. 3.4 and the uniqueness of Z is studied in Sec. 3.5.

3.1. v-Tangles and w-Tangles. Building on Section 2.4, we define v-tangles and w-tangles combinatorially as finitely presented circuit algebras, given by generators and relations. This facilitates the algebraic treatment we present, by which we connect w-tangle theory to Kashiwara-Vergne theory. In Section 3.4 we recall Satoh’s tubing map [WKO1, Sec. 3.1.1], to endow w-tangles with the expected topological meaning.

Definition 3.1. The (S-graded) circuit algebra vD of v-tangle diagrams is the S-graded directed circuit algebra freely generated by two generators in C_{2,2} called the positive crossing, � _{1 2}^{4 3}, and the negative crossing, � _{1 2}^{4 3}. In as much as possible we suppress the leg-numebering below; with this in mind, vD :=^{CA} , . The skeleton of both crossings is the element

� _{1 2}^{4 3} (the pairing of 1&3 and 2&4) in S_{2,2}. That is, ς(�) = ς(�) = �.

Example 3.2. An example of a v-tangle diagram V is shown the left side of Figure 2. V is a circuit algebra composition of two negative crossings and one positive crossing by the wiring diagram D, as shown. The right side of the same figure shows the skeleton ς(V) of V: to produce the skeleton, replace each crossing by the element � in S and apply the same wiring diagram. The elements of S are oriented 1-manifolds with numbered boundary points, and hence the result is equal to the one shown in the figure.

Warning 3.3. People familiar with the planar presentation of virtual tangles may be accustomed to the notion of there being another type of crossing: the “virtual crossing”. The main point of introducing circuit algebras (as opposed to working with planar algebras) is to eliminate the need for virtual crossings: they become part of the CA structure. This greatly simplifies the presentation of both v- and w-tangles: there is one less generator, as seen above, and far fewer relations, as we explain in Remark 3.5.

Definition 3.4. The (S-graded) circuit algebra vT of v-tangles is the S-graded directed circuit algebra of v-tangle diagrams vD, modulo the R1^{s}, R2 and R3 moves as depicted in Figure 3. These relations make sense as circuit algebra relations between the two generators, and preserve skeleta. To obtain the circuit algebra wT of w-tangles we also mod out by the OC relation of Figure 3 (note that each side in that relation involves only two generators, with the apparent third “virtual” crossing being merely a circuit algebra artifact). In fewer

words, vT :=^{CA} ,  =  = , = , and wT :=^{vT}  _{=} .

Remark 3.5. One may also define v-tangles and w-tangles using the language of planar algebras, except then another generator is required (the “virtual crossing”) and also a number of further relations shown in Figure 3 (VR1–VR3, M), and some of the operations (non-planar wirings) become less elegant to define. In our context “virtual crossings” are automatically present (but unimportant) as part of the circuit algebra structure, and the “virtual Reidemeister moves” VR1–VR3 and M are also automatically true. In fact, the “rerouting move” known in the planar presentation, which says that a purely virtual strand of a v-tangle diagram can be re-routed in any other purely virtual way, is precisely the statement that virtual crossings are unimportant, and the language of circuit algebras makes this fact manifest.

Remark 3.6. For S ∈ S a given skeleton, that is, an oriented 1-manifold with numbered ends, let us denote by vT (S) and wT (S), respectively, the v- and w-tangles with skeleton S. That is, vT (S) and wT (S) are the pre-images of S under the skeleton map ς. Note that in our case the skeleton map is “forgetting topology”, in other words, forgetting the under/over information of crossings, resulting in empty circuits. With this notation, wT (↑), the set of

1 ^{2}  ^{3}  ^{4}  ^{5} D = V =  ^{6}^{1}^{2}  ^{3}  ^{5}^{4} ^{6}  ^{1}^{2}  ^{3}  ^{5}^{4} ^{6}  ς(V) =^{3}^{1}^{2}^{1} ^{3}^{4} ^{1} ^{4}^{2} ^{4} ^{3}^{2}

Figure 2. V ∈ vD_{3,3} is a v-tangle diagram. V is the result of applying the circuit algebra operation D : C_{2,2} × C_{2,2} × C_{2,2} → C_{3,3}, given by the wiring diagram shown, acting on two negative crossings and one positive crossing. In other words V = D(�, �, �). The skeleton of V is given by ς(V) = D(�, �, �), which is equal in S to the diagram shown here. Note that we usually suppress the circuit algebra numbering of boundary points. Note also that the apparent “virtual crossings” of V are not virtual crossings but merely part of the circuit algebra structure, see Warning 3.3. The same is true for the crossings appearing in the skeleton ς(V).

= VR2 = VR1 = R3 ≠ R1  = R2 =  =^{w}  ≠ M  OC  UC  = VR3 R1^{s} =

Figure 3. The relations (“Reidemeister moves”) R1^{s}, R2 and R3 define v-tangles, adding OC to these defines w-tangles. VR1, VR2, VR3 and M are not necessary as the circuit algebra presentation eliminates the need for “virtual crossings” as generators. R1 is not imposed for framing reasons, and not imposing UC breaks the symmetry between over and under crossings in wT.

w-tangles whose skeleton is a single line, is exactly the set of (long) w-knots discussed in [WKO1, Section 3]. Note also that wT (↑_{n}), the set of w-tangles whose skeleton is n lines, includes w-braids with n strands ([WKO1, Section 2]) but it is more general. Neither w-knots nor w-braids are circuit algebras.

Remark 3.7. Since we do not mod out by the R1 relation, only by its weak (or “spun”) version R1^{s}, it is more appropriate to call our class of v/w-tangles framed v/w-tangles. (Recall that framed u-tangles are characterized as the planar algebra generated by the positive and negative crossings modulo the R1^{s}, R2 and R3 relations.) However, since we are for the most part interested in studying the framed theories (cf. Comment 4.3), we will reserve the unqualified name for the framed case, and will explicitly write “unframed v/w-tangles” if we wish to mod out by R1. For a more detailed explanation of framings and R1 moves, see [WKO1, Remark 3.5].

Our next task is to study the associated graded structures grad vT and grad wT of vT and wT, with respect to the augmentation ideal as described in Section 2.2. These are “arrow diagram spaces on tangle skeletons”: directed analogues of the chord diagram spaces of ordinary finite type invariant theory, and even more similar to the arrow diagram spaces for braids and knots discussed in [WKO1]. Our convention for figures will be to show skeletons as thick lines with thin arrows (directed chords). Again, the language of circuit algebras makes defining these spaces exceedingly simple.

π  −

Definition 3.8. The (S-graded) circuit algebra D^{v} = D^{w} of arrow diagrams is the graded and S-graded directed circuit algebra generated by a single degree 1 generator a in C_{2,2} called “the arrow” as shown on the right, with the obvious meaning for its skeleton. There are morphisms π: D^{v} → vT and π: D^{w} → wT defined by mapping the arrow to an overcrossing minus a no-crossing. (On the right some virtual crossings were added to make the skeleta match). Let A^{v} be D^{v}/6T, let A^{w} := A^{v}/T C = D^{w}/(\overrightarrow{4T}, T C), and let A^{sv} := A^{v}/RI and A^{sw} := A^{w}/RI, with RI, 6T, \overrightarrow{4T}, and T C being the relations shown in Figures 4 and 5. Note that the pair of relations (\overrightarrow{4T}, T C) is equivalent to the pair (6T, T C), as discussed in [WKO1, Section 2.3.1].

+  + + + =^{6T}  =^{RI}

Figure 4. Relations for v-arrow diagrams on tangle skeletons. Skeleta parts that are not connected can lie on separate skeleton components; and the dotted arrow that remains in the same position means “all other arrows remain the same throughout”.

and + =^{\overrightarrow{4T}}  =^{TC} +

Figure 5. Relations for w-arrow diagrams on tangle skeletons.

Proposition 3.9. The maps π above induce surjections π: A^{sv} → grad vT and π: A^{sw} → grad wT. Hence in the language of Definition 2.6, A^{sv} and A^{sw} are candidate associated graded structures of vT and wT.

Proof. Proving that π is well-defined amounts to checking directly that the RI and 6T or RI, \overrightarrow{4T} and TC relations are in the kernel of π. (Just like in the finite type theory of virtual knots and braids.) Thanks to the circuit algebra structure, it is enough to verify the surjectivity of π in degree 1. We leave this as an exercise for the reader. □

We do not know if A^{sv} is indeed the associated graded of vT (also see [BHLR]). Yet in the w case, the picture is simple:

Theorem 3.10. The assignment � ↦ e^{a} (with e^{a} denoting the exponential of a single arrow from the over strand to the under strand, interpreted via its power series) extends to a well defined Z: wT → A^{sw}. The resulting map Z is a homomorphic A^{sw}-expansion, and in particular, A^{sw} ≅ grad wT and Z is a homomorphic expansion.

Proof. The proof is essentially the same as the proof of [WKO1, Theorem 2.15], and follows [BP, AT]. One needs to check that Z satisfies the Reidemeister moves and the OC relation. R1^{s} follows easily from RI, R2 is obvious, TC implies OC. For R3, let A^{sw}(↑_{n}) denote the space of “arrow diagrams on n vertical strands”. We need to verify that R := e^{a} ∈ A^{sw}(↑_{2}) satisfies the Yang-Baxter equation

R^{12}R^{13}R^{23} = R^{23}R^{13}R^{12}, in A^{sw}(↑_{3}),

\overrightarrow{ST U}_{1}:  e  =  e  −  e  \overrightarrow{ST U}_{2}:  e  =  e  −  e

Figure 6. The \overrightarrow{ST U} relations for arrow diagrams, with their “central edges” marked e for easier memorization.

where R^{ij} = e^{a_{ij}} means “place R on strands i and j”. By 4T and T C relations, both sides of the equation can be reduced to e^{a_{12}+a_{13}+a_{23}}, proving the Reidemeister invariance of Z.

Z is by definition a circuit algebra homomorphism. Hence to show that Z is an A^{sw}- expansion we only need to check the universality property in degree one, where it is very easy. The rest follows from Proposition 2.7. □

Remark 3.11. Note that the restriction of Z to w-knots and w-braids (in the sense of Remark 3.6) recovers the expansions constructed in [WKO1]. Note also that the filtration and associated graded structure for w-braids fits into the general algebraic framework of Section 2 by applying the machinery to the skeleton-graded group of w-braids instead the circuit algebra of w-tangles. (The skeleton of a w-braid is the permutation it represents.) However, as w-knots do not form a finitely presented algebraic structure in the sense of Section 2, the “finite type” filtration used in [WKO1] does not arise as powers of any augmentation ideal. This captures the reason why w-knots are “the wrong objects to study”, as we have mentioned at the beginning of Section 3 of [WKO1].

In a similar spirit to [WKO1, Definition 3.12], one may define a “w-Jacobi diagram” on an arbitrary skeleton:

Definition 3.12. A “w-Jacobi diagram on a tangle skeleton”^{9} is a graph made of the following ingredients:

• An oriented “skeleton” consisting of long lines and circles (i.e., an oriented onemanifold). In figures we draw the skeleton lines thicker.

• Other directed edges, usually called “arrows”.

• Trivalent “skeleton vertices” in which an arrow starts or ends on the skeleton line.

• Trivalent “internal vertices” in which two arrows end and one arrow begins. The internal vertices are cyclically oriented; in figures the assumed orientation is always counterclockwise unless marked otherwise. Furthermore, all trivalent vertices must be connected to the skeleton via arrows (but not necessarily following the direction of the arrows).

Note that we allow multiple and loop arrow edges, as long as trivalence and the two-in-one-out rule is respected.

Formal linear combinations of (w-Jacobi) arrow diagrams form a circuit algebra. We denote by A^{wt} the quotient of the circuit algebra of arrow diagrams modulo the \overrightarrow{ST U}_{1}, \overrightarrow{ST U}_{2} relations of Figure 6, and the TC relation. We denote A^{wt} modulo the RI relation by A^{swt}. We then have the following “bracket-rise” theorem:

9We usually short this to “w-Jacobi diagram”, or sometimes “arrow diagram” or just “diagram”.

\overrightarrow{AS}:  0 =  ^{+}  ^{\overrightarrow{IHX}:}  ^{e}  ^{=}  ^{e}  ^{−}  ^{e}
Figure 7. The \overrightarrow{AS} and \overrightarrow{IHX} relations.

=  −^{\overrightarrow{ST U}_{1}}^{\overrightarrow{ST U}_{2}} e_{2}  e_{1}

Figure 8. Applying \overrightarrow{ST U}_{1} and \overrightarrow{ST U}_{2} to the diagram on the left, we get the two sides of \overrightarrow{4T}.

RI:  = 0

Figure 9. A 4-wheel and the RI relation re-phrased.

Theorem 3.13. The obvious inclusion of arrow diagrams (with no internal vertices) into w-Jacobi diagrams descends to a map ῑ: A^{w} → A^{wt}, which is a circuit algebra isomorphism. Furthermore, the \overrightarrow{AS} and \overrightarrow{IHX} relations of Figure 7 hold in A^{wt}. Consequently, it is also true that A^{sw} ≅ A^{swt}.

Proof. In the proof of [WKO1, Theorem 3.14] we showed this for long w-knots (i.e., tangles whose skeleton is a single long line). That proof applies here verbatim, noting that it does not make use of the connectivity of the skeleton.

In short, to check that ῑ is well-defined, we need to show that the \overrightarrow{ST U} relations imply the \overrightarrow{4T} relation. This is shown in Figure 8. To show that ῑ is an isomorphism, we construct an inverse A^{wt} → A^{w}, which “eliminates all internal vertices” using a sequence of \overrightarrow{ST U} relations. Checking that this is well-defined requires some case analysis; the fact that it is an inverse to ῑ is obvious. Verifying that the \overrightarrow{AS} and \overrightarrow{IHX} relations hold in A^{wt} is an easy exercise. □

Given the above theorem, we no longer keep the distinction between A^{w} and A^{wt} and between A^{sw} and A^{swt}.

We recall from [WKO1] that a “k-wheel”, sometimes denoted w_{k}, is a an arrow diagram consisting of an oriented cycle of arrows with k incoming “spokes”, the tails of which rest on the skeleton. An example is shown in Figure 9. In this language, the RI relation can be rephrased using the \overrightarrow{ST U} relation to say that all one-wheels are 0, or w_{1} = 0.

Remark 3.14. Note that if T is an arbitrary w tangle, then the equality on the left side of the figure below always holds, while the one on the right generally doesn’t:

T  =  T ,  yet  T  ≠  T .  (2)

The arrow diagram version of this statement is that if D is an arbitrary arrow diagram in A^{w}, then the left side equality in the figure below always holds (we will sometimes refer to this as the “head-invariance” of arrow diagrams), while the right side equality (“tail-invariance”) generally fails.

yet = 0, D +  ≠ 0. D +  (3)

We leave it to the reader to ascertain that Equation (2) implies Equation (3). There is also a direct proof of Equation (3) which we also leave to the reader, though see an analogous statement and proof in [BN2, Lemma 3.4]. Finally note that a restricted version of tailinvariance does hold — see Section 3.3.

3.2. A^{w}(↑_{n}) and the Alekseev-Torossian Spaces.

Definition 3.15. Let A^{v}(↑_{n}) be the part of A^{v} in which the skeleton is the disjoint union of n directed lines, with similar definitions for A^{w}(↑_{n}), A^{sv}(↑_{n}), and A^{sw}(↑_{n}).

Theorem 3.16. (Diagrammatic PBW Theorem.) Let B_{n}^{w} denote the space of uni-trivalent diagrams^{10} with symmetrized ends coloured with colours in some n-element set (say {x_{1}, …, x_{n}}), modulo the \overrightarrow{AS} and \overrightarrow{IHX} relations of Figure 7. Then there is an isomorphism A^{w}(↑_{n}) ≅ B_{n}^{w}.

Proof sketch. Readers familiar with the diagrammatic PBW theorem [BN1, Theorem 8] will note that the proof carries through almost verbatim. There is a map χ : B_{n}^{w} → A^{w}(↑_{n}), which sends each uni-trivalent diagram to the average of all ways of attaching their univalent ends to the skeleton of n lines, so that ends of colour x_{i} are attached to the strand numbered i. I.e., a diagram with k_{i} uni-valent vertices of colour x_{i} is sent to a sum of ∏_{i} k_{i}! terms, divided by ∏_{i} k_{i}!.

The goal is to show that χ is an isomorphism by constructing an inverse for it. The image of χ are symmetric sums of diagrams, that is, sums of diagrams that are invariant under permuting arrow endings on the same skeleton component. One can show that in fact any arrow diagram D in A^{w}(↑_{n}) is equivalent via \overrightarrow{ST U} and T C relations to a symmetric sum. The obvious candidate is its “symmetrization” Sym(D): the average of all ways of permuting the arrow endings on each skeleton component of D. It is not true that each diagram is equivalent to its symmetrization (hence, the “simply delete the skeleton” map is not an inverse for χ), but it is true that D − Sym(D) has fewer skeleton vertices (lower degree) than D, hence we can construct χ^{−1} inductively. The fact that this inductive procedure is well-defined requires a proof; that proof is essentially the same as the proof of the corresponding fact in [BN1, Theorem 8]. □

10Oriented graphs with vertex degrees either 1 or 3, where trivalent vertices must have two edges incoming and one edge outgoing and are cyclically oriented.

x_{1} x_{1}  x_{2}  x_{1} x_{1} x_{3} apply \overrightarrow{IHX} here first

Figure 10. A wheel of trees can be reduced to a combination of wheels, and a wheel of trees with a Little Prince.

Both A^{w}(↑_{n}) and B_{n}^{w} have a natural bi-algebra structure. In A^{w}(↑_{n}) multiplication is given by stacking. For a diagram D ∈ A^{w}(↑_{n}), the co-product Δ(D) is given by the sum of all ways of dividing D between a “left co-factor” and a “right cofactor” so that the connected components of D − S are kept intact, where S is the skeleton of D. In B_{n}^{w} multiplication is given by disjoint union, and Δ is the sum of all ways of dividing the connected components of a diagram between two co-factors (here there is no skeleton). Note that the isomorphism χ above is a co-algebra isomorphism, but not an algebra homomorphism.

The primitives P_{n}^{w} of B_{n}^{w} are the connected diagrams (and hence the primitives of A^{w}(↑_{n}) are the diagrams that remain connected even when the skeleton is removed). Given the “two in one out” rule for internal vertices, the diagrams in P_{n}^{w} can only be trees (diagrams with no cycles) or wheels (a single oriented cycle with a number of “spokes”, or leaves, attached to it). “Wheels of trees” can be reduced to simple wheels by repeatedly using \overrightarrow{IHX}, as in Figure 10.

Thus as a vector space P_{n}^{w} is easy to identify. It is a direct sum P_{n}^{w} = ⟨trees⟩ ⊕ ⟨wheels⟩. The wheels part is simply the graded vector space generated by all cyclic words in the letters x_{1}, …, x_{n}. Alekseev and Torossian [AT] denote the space of cyclic words by 𝔱𝔯_{n}, and so shall we. The trees in P_{n}^{w} have leafs coloured x_{1}, …, x_{n}. Modulo \overrightarrow{AS} and \overrightarrow{IHX}, they correspond to elements of the free Lie algebra 𝔩𝔦𝔢_{n} on the generators x_{1}, …, x_{n}. But the root of each such tree also carries a label in {x_{1}, …, x_{n}}, hence there are n types of such trees as separated by their roots, and so P_{n}^{w} is linearly isomorphic to the direct sum 𝔱𝔯_{n} ⊕ ⨁_{i=1}^{n} 𝔩𝔦𝔢_{n}.

Note that with B_{n}^{sw} and P_{n}^{sw} defined in the analogous manner (i.e., factoring out by one-wheels, as in the RI relation), we can also conclude that there is a linear isomorphism P_{n}^{sw} ≅ 𝔱𝔯_{n} /(deg 1) ⊕ ⨁_{i=1}^{n} 𝔩𝔦𝔢_{n}.

By the Milnor-Moore theorem [MM], A^{w}(↑_{n}) is isomorphic to the universal enveloping algebra U(P_{n}^{w}), with P_{n}^{w} identified as the subspace P^{w}(↑_{n}) of primitives of A^{w}(↑_{n}) using the PBW symmetrization map χ: B_{n}^{w} → A^{w}(↑_{n}). Thus in order to understand A^{w}(↑_{n}) as an associative algebra, it is enough to understand the Lie algebra structure induced on P_{n}^{w} via the commutator bracket of A^{w}(↑_{n}).

Our goal is to identify P^{w}(↑_{n}) as the Lie algebra 𝔱𝔯_{n} ⋊(𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n}), which in itself is a combination of the Lie algebras 𝔞_{n}, 𝔱𝔡𝔢𝔯_{n} and 𝔱𝔯_{n} studied by Alekseev and Torossian [AT]. Here are the relevant definitions:

Definition 3.17. Let 𝔞_{n} denote the vector space with basis x_{1}, …, x_{n}, also regarded as an Abelian Lie algebra of dimension n. As before, let 𝔩𝔦𝔢_{n} = 𝔩𝔦𝔢(𝔞_{n}) denote the free Lie algebra on n generators, now identified as the basis elements of 𝔞_{n}. Let 𝔡𝔢𝔯_{n} = 𝔡𝔢𝔯(𝔩𝔦𝔢_{n}) be the

1  ^{2}  ^{3} ⟼^{π} (0, [[x_{1}, x_{2}], [x_{2}, x_{3}]], 0)

Figure 11. The map π.

(graded) Lie algebra of derivations acting on 𝔩𝔦𝔢_{n}, and let

𝔱𝔡𝔢𝔯_{n} = {D ∈ 𝔡𝔢𝔯_{n}: ∀i ∃a_{i} s.t. D(x_{i}) = [x_{i}, a_{i}]}

denote the subalgebra of “tangential derivations”. A tangential derivation D is determined by the a_{i}’s for which D(x_{i}) = [x_{i}, a_{i}], and determines them up to the ambiguity a_{i} ↦ a_{i}+α_{i}x_{i}, where the α_{i}’s are scalars. Thus as vector spaces, 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n} ≅ ⨁_{i=1}^{n} 𝔩𝔦𝔢_{n}.

Definition 3.18. Let Ass_{n} = U(𝔩𝔦𝔢_{n}) be the free associative algebra “of words”, and let Ass_{n}^{+} be the degree > 0 part of Ass_{n}. As before, we let 𝔱𝔯_{n} = Ass_{n}^{+} /(x_{i_{1}}x_{i_{2}} ⋯ x_{i_{m}} = x_{i_{2}} ⋯ x_{i_{m}}x_{i_{1}}) denote “cyclic words” or “(coloured) wheels”. Ass_{n}, Ass_{n}^{+}, and 𝔱𝔯_{n} are 𝔱𝔡𝔢𝔯_{n}-modules and there is an obvious equivariant “trace” tr: Ass_{n}^{+} → 𝔱𝔯_{n}.

Proposition 3.19. There is a split short exact sequence of Lie algebras

0 ⟶ 𝔱𝔯_{n} \underrightarrow{ι} P^{w}(↑_{n}) \underrightarrow{π} 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n} ⟶ 0.

Proof. The inclusion ι is defined the natural way: 𝔱𝔯_{n} is spanned by coloured “floating” wheels, and such a wheel is mapped into P^{w}(↑_{n}) by attaching its ends to their assigned strands in arbitrary order. Note that this is well-defined: wheels have only tails, and tails commute.

As vector spaces, the statement is already proven: P^{w}(↑_{n}) is generated by trees and wheels (with the all arrow endings fixed on n strands). When factoring out by the wheels, only trees remain. Trees have one head and many tails. All the tails commute with each other, and commuting a tail with a head on a strand costs a wheel (by \overrightarrow{ST U}), thus in the quotient the head also commutes with the tails. Therefore, the quotient is the space of coloured “floating” trees, which we have previously identified with ⨁_{i=1}^{n} 𝔩𝔦𝔢_{n} ≅ 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n}.

It remains to show that the maps ι and π are Lie algebra maps as well. For ι this is easy: the Lie algebra 𝔱𝔯_{n} is commutative, and is mapped to the commutative (due to T C) subalgebra of P^{w}(↑_{n}) generated by wheels. Next, we show that π is a homomorphism. The map π quotients out by wheels and reads trees as lie words in 𝔩𝔦𝔢_{n}^{⊕n}, as show in Figure 11 and below:

π : P^{w}(↑_{n}) → 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n} ≅^{v.s.}  𝔩𝔦𝔢_{n}^{⊕n}
⟨wheels⟩ ∋ w ↦  0
⟨trees⟩ ∋ T ↦  (0, …, a_{i}, …, 0)

Here a_{i} is the lie word corresponding to T in the i-th component of 𝔩𝔦𝔢_{n}^{⊕n}, where the head of T is on strand i. Namely, a_{i} is a lie word on the generators corresponding to the strand numbers to which the tails of T are attached, and commutators corresponding to each of the arrow vertex of T, read left to right when looking at T from its head.

To show that π is a map of Lie algebras we give two proofs, first a “hands-on” one, then a “conceptual” one.

Hands-on argument. 𝔞_{n} is the image of single arrows on one strand. These commute with everything in P^{w}(↑_{n}), and so does 𝔞_{n} in the direct sum 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n}. Thus, π respects commutators involving these local arrows.

It remains to show that commuting trees in P^{w}(↑_{n}) maps to the bracket of 𝔱𝔡𝔢𝔯_{n}; or more accurately, the bracket of 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n} transferred to 𝔩𝔦𝔢_{n}. Let D and D′ be elements of 𝔱𝔡𝔢𝔯_{n} represented by (a_{1}, …, a_{n}) and (a′_{1}, …, a′_{n}), meaning that D(x_{i}) = [x_{i}, a_{i}] and D′(x_{i}) = [x_{i}, a′_{i}] for i = 1, …, n. We compute the commutator of these elements:

[D, D′](x_{i}) = (DD′ − D′D)(x_{i}) = D[x_{i}, a′_{i}] − D′[x_{i}, a_{i}] = = [[x_{i}, a_{i}], a′_{i}] + [x_{i}, Da′_{i}] − [[x_{i}, a′_{i}], a_{i}] − [x_{i}, D′a_{i}] = [x_{i}, Da′_{i} − D′a_{i} + [a_{i}, a′_{i}]].

Now let T and T′ be two trees in P^{w}(↑_{n})/ 𝔱𝔯_{n}, with heads on strands i and j, respectively (i may or may not equal j). Let us denote by a_{i} (resp. a′_{j}) the element in 𝔩𝔦𝔢_{n} corresponding to T (resp. T′), as above. In 𝔱𝔡𝔢𝔯_{n}, let D = π(T) = (0, …, −a_{i}, …, 0) and D′ = π(T′) = (0, …, −a_{j}, …, 0). (In each case, the i-th, respectively j-th, is the only non-zero component.) The commutator of these elements is given by [D, D′](x_{i}) = [Da′_{i} − D′a_{i} + [a_{i}, a′_{i}], x_{i}], and [D, D′](x_{j}) = [Da′_{j} − D′a_{j} + [a_{j}, a′_{j}], x_{j}]. Note that unless i = j, a_{j} = a′_{i} = 0.

In P^{w}(↑_{n})/ 𝔱𝔯_{n} tails commute, as well as the head of a tree with its own tails. Therefore, commuting two trees only incurs a cost when commuting a head of one tree over the tails of the other on the same strand, and the two heads over each other, if i = j.

If i ≠ j, then commuting the head of T over the tails of T′ by \overrightarrow{ST U} gives a sum of trees given by −Da′_{j}, with heads on strand j, while moving the head of T′ over the tails of T costs exactly D′a_{i}, with heads on strand i, as needed.

If i = j, then everything happens on strand i, and the cost is (−Da′_{i} + D′a_{i} − [a_{i}, a′_{i}]), where the last term arises from commuting the two heads.

Conceptual argument. There is an action of P^{w}(↑_{n}) on 𝔩𝔦𝔢_{n}, as follows: introduce and extra strand on the right. An element L of 𝔩𝔦𝔢_{n} corresponds to a tree with its head on the extra strand. Its commutator with an element of P^{w}(↑_{n}) (considered as an element of P^{w}(↑_{n+1}) by the obvious inclusion) is again a tree with head on strand (n + 1), defined to be the result of the action.

Since L has only tails on the first n strands, elements of 𝔱𝔯_{n}, which also only have tails, act trivially. So do single (local) arrows on one strand (𝔞_{n}). It remains to show that trees act as 𝔱𝔡𝔢𝔯_{n}, and it is enough to check this on the generators of 𝔩𝔦𝔢_{n} (as the Leibniz rule is obviously satisfied). The generators of 𝔩𝔦𝔢_{n} are arrows pointing from one of the first n strands, say strand i, to strand (n + 1). A tree T with head on strand i acts on this element, according \overrightarrow{ST U}, by forming the commutator [x_{i}, T], which is exactly the action of 𝔱𝔡𝔢𝔯_{n}.

To identify P^{w}(↑_{n}) as the semidirect product 𝔱𝔯_{n} ⋊(𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n}), it remains to show that the short exact sequence of the Proposition splits. This is indeed the case, although not canonically. Two —of the many— splitting maps u, l: 𝔱𝔡𝔢𝔯_{n} ⊕𝔞_{n} → P^{w}(↑_{n}) are described as follows: 𝔱𝔡𝔢𝔯_{n} ⊕𝔞_{n} is identified with ⨁_{i=1}^{n} 𝔩𝔦𝔢_{n}, which in turn is identified with “floating” coloured trees. A map to P^{w}(↑_{n}) can be given by specifying how to place the legs on their specified strands. A tree may have many tails but has only one head, and due to T C, only the positioning of the head matters. Let u (for upper) be the map placing the head of each tree above all its tails on the same strand, while l (for lower) places the head below all the tails. It is clear that these are both Lie algebra maps and that π ∘ u and π ∘ l are both the identity of 𝔱𝔡𝔢𝔯_{n} ⊕𝔞_{n}. This makes P^{w}(↑_{n}) a semidirect product. □

Remark 3.20. Let 𝔱𝔯_{n}^{s} denote 𝔱𝔯_{n} mod out by its degree one part (one-wheels). Since the RI relation is in the kernel of π, there is a similar split exact sequence

0 → 𝔱𝔯_{n}^{s} →^{\overline{ι}} P^{sw} →^{\overline{π}} 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n} .

Definition 3.21. For any D ∈ 𝔱𝔡𝔢𝔯_{n}, (l − u)D is in the kernel of π, therefore is in the image of ι, so ι^{−1}(l − u)D makes sense. We call this element divD.

Definition 3.22. In [AT] div is defined as follows: div(a_{1}, …, a_{n}) := ∑_{k=1}^{n} tr((∂_{k}a_{k})x_{k}), where ∂_{k} picks out the words of a sum which end in x_{k} and deletes their last letter x_{k}, and deletes all other words (the ones which do not end in x_{k}).

Proposition 3.23. The div of Definition 3.21 and the div of [AT] are the same.

... x_{i_{2}} x_{i_{1}}  x^{i_{k}} x_{i_{k−1}}

Proof. It is enough to verify the claim for the linear generators of 𝔱𝔡𝔢𝔯_{n}, namely, elements of the form (0, …, a_{j}, …, 0), where a_{j} ∈ 𝔩𝔦𝔢_{n} or equivalently, single (floating, coloured) trees, where the colour of the head is j. By the Jacobi identity, each a_{j} can be written in a form a_{j} = [x_{i_{1}}, [x_{i_{2}}, […, x_{i_{k}}] …]. Equivalently, by \overrightarrow{IHX}, each tree has a standard “comb” form, as shown on the picture on the right.

For an associative word Y = y_{1}y_{2} … y_{l} ∈ Ass_{n}^{+}, we introduce the notation

[Y] := [y_{1}, [y_{2}, […, y_{l}] …]. The div of [AT] picks out the words that end in x_{j}, forgets the rest, and considers these as cyclic words. Therefore, by interpreting the Lie brackets as commutators, one can easily check that for a_{j} written as above,

div((0, …, a_{j}, …, 0)) = ∑_{α: i_{α}=x_{j}} −x_{i_{1}} … x_{i_{α−1}}[x_{i_{α+1}} … x_{i_{k}}]x_{j}.  (4)

j  − j  = j

In Definition 3.21, div of a tree is the difference between attaching its head on the appropriate strand (here, strand j) below all of its tails and above. As shown in the figure on the right, moving the head across each of the tails on strand j requires an \overrightarrow{ST U} relation, which “costs” a wheel (of trees, which is equivalent to a sum of honest wheels). Namely, the head gets connected to the tail in question. So div of the tree represented by a_{j} is given by

∑_{α: x_{iα}=j}“connect the head to the α leaf”.

This in turn gets mapped to the formula above via the correspondence between wheels and cyclic words. □

−  =

Remark 3.24. There is an action of 𝔱𝔡𝔢𝔯_{n} on 𝔱𝔯_{n} as follows. Represent a cyclic word w ∈ 𝔱𝔯_{n} as a wheel in P^{w}(↑_{n}) via the map ι. Given an element D ∈ 𝔱𝔡𝔢𝔯_{n}, u(D), as defined above, is a tree in P^{w}(↑_{n}) whose head is above all of its tails. We define D · w := ι^{−1}(u(D)ι(w) − ι(w)u(D)). Note that u(D)ι(w) − ι(w)u(D) is in the image of ι, i.e., a linear combination of wheels, for the following reason. The wheel ι(w) has only tails. As we commute the tree u(D) across the wheel, the head of the tree is commuted across tails of the wheel on the same strand. Each time this happens the cost, by the \overrightarrow{ST U} relation, is a wheel with the tree attached to it, as shown on the right, which in turn (by \overrightarrow{IHX} relations, as Figure 10 shows) is a sum of wheels. Once the head of the tree has been moved to the top, the tails of the tree commute up for free by T C. Note that the alternative definition, D · w := ι^{−1}(l(D)ι(w) − ι(w)l(D)) is in fact equal to the definition above.

Definition 3.25. In [AT], the group TAut_{n} is defined as exp(𝔱𝔡𝔢𝔯_{n}). Note that 𝔱𝔡𝔢𝔯_{n} is positively graded, hence it integrates to a group. Note also that TAut_{n} is the group of “basis-conjugating” automorphisms of 𝔩𝔦𝔢_{n}, i.e., for g ∈ TAut_{n}, and any x_{i}, i = 1, …, n generator of 𝔩𝔦𝔢_{n}, there exists an element g_{i} ∈ exp(𝔩𝔦𝔢_{n}) such that g(x_{i}) = g_{i}^{−1}x_{i}g_{i}.

Note that the group multiplication in TAut_{n} is the one exponentiated from 𝔱𝔡𝔢𝔯_{n}, which is read left to right (as right actions) rather than right to left (as function composition). For example, for f, g ∈ TAut_{n}, (fg)(x_{i}) = g(f(x_{i})) = g(f_{i}^{−1}x_{i}f_{i}) = g(f_{i})^{−1}g_{i}^{−1}x_{i}g_{i}g(f_{i}).

The action of 𝔱𝔡𝔢𝔯_{n} on 𝔱𝔯_{n} lifts to an action of TAut_{n} on 𝔱𝔯_{n}, by interpreting exponentials formally, in other words e^{D} acts as ∑_{n=0}^{∞} \frac{D^{n}}{n!} . The lifted action is by conjugation: for w ∈ 𝔱𝔯_{n} and e^{D} ∈ TAut_{n}, e^{D} · w = ι^{−1}(e^{uD}ι(w)e^{−uD}).

Recall that in Section 5.1 of [AT] Alekseev and Torossian construct a map j: TAut_{n} → 𝔱𝔯_{n} which is characterized by two properties: the cocycle property

j(gh) = j(g) + g · j(h),  (5)

where in the second term multiplication by g denotes the action described above; and the condition

\frac{d}{ds}j(exp(sD))|_{s=0} = div(D).  (6)

Now let us interpret j in our context.

Definition 3.26. The adjoint map ∗: A^{w}(↑_{n}) → A^{w}(↑_{n}) acts by “flipping over diagrams and negating arrow heads on the skeleton”. In other words, for an arrow diagram D,

D^{∗} := (−1)^{#{tails on skeleton}}S(D),

where S denotes the map which switches the orientation of the skeleton strands (i.e. flips the diagram over), and multiplies by (−1)^{#skeleton vertices}.

Note that the number of tails on the skeleton is the same as the degree of the arrow diagram, hence D^{∗} = (−1)^{deg D}S(D).

Proposition 3.27. For D ∈ 𝔱𝔡𝔢𝔯_{n}, define a map J: TAut_{n} → exp(𝔱𝔯_{n}) by J(e^{D}) := e^{uD}(e^{uD})^{∗}. Then

exp(j(e^{D})) = J(e^{D}).

Proof. Note that (e^{uD})^{∗} = e^{−lD}, due to “Tails Commute” and the fact that a tree has only one head.

Let us check that log J satisfies properties (5) and (6). Namely, with g = e^{D_{1}} and h = e^{D_{2}}, and using that 𝔱𝔯_{n} is commutative, we need to show that

J(e^{D_{1}}e^{D_{2}}) = J(e^{D_{1}})(e^{uD_{1}} · J(e^{D_{2}})),  (7)

where · denotes the action of 𝔱𝔡𝔢𝔯_{n} on 𝔱𝔯_{n}; and that

\frac{d}{ds}J(e^{sD})|_{s=0} = div D.  (8)

Indeed, with BCH(D_{1}, D_{2}) = log e^{D_{1}}e^{D_{2}} being the standard Baker–Campbell–Hausdorff formula,

J(e^{D_{1}}e^{D_{2}}) = J(e^{BCH(D_{1},D_{2})}) = e^{u(BCH(D_{1},D_{2})}e^{−l(BCH(D_{1},D_{2})} = e^{BCH(uD_{1},uD_{2})}e^{− BCH(lD_{1},lD_{2})}

= e^{uD_{1}}e^{uD_{2}}e^{−lD_{2}}e^{−lD_{1}} = e^{uD_{1}}(e^{uD_{2}}e^{−lD_{2}})e^{−uD_{1}}e^{uD_{1}}e^{lD_{1}} = (e^{uD_{1}} · J(D_{2}))J(D_{1}),

as needed.

As for condition (6), a direct computation of the derivative yields

\frac{d}{ds}J(e^{sD})|_{s=0} = uD − lD = div D,

as desired. □

3.3. The Relationship with u-Tangles. Let uT be the planar algebra of classical, or “usual” tangles. There is a map a: uT → wT of u-tangles into w-tangles: algebraically, it is defined in the obvious way on the planar algebra generators of uT. (It can also be interpreted topologically as Satoh’s tubing map, see [WKO1, Section 3.1.1], where a u-tangle is a tangle drawn on a sphere. However, it is only conjectured that the circuit algebra presented here is a Reidemeister theory for “tangled ribbon tubes in ℝ^{4}”.) The map a induces a corresponding map α: A^{u} → A^{sw}, which maps an ordinary Jacobi diagram (i.e., unoriented chords with internal trivalent vertices modulo the usual AS, IHX and ST U relations) to the sum of all possible orientations of its chords (many of which are zero in A^{sw} due to the “two in one out” rule).

\xymatrix{uT \ar@{.>}[r]^{Z^{u}} \ar[d]^{a} & A^{u} \ar[d]^{α} \\ wT \ar[r]^{Z^{w}} & A^{sw}}

It is tempting to ask whether the square on the left commutes. Unfortunately, this question hardly makes sense, as there is no canonical choice for the dotted line in it. Similarly to the braid case of [WKO1, Section 2.5.5], the definition of the homomorphic expansion (Kontsevich integral) for utangles typically depends on various choices of “parenthesizations”. Choos-

ing parenthesizations, this square becomes commutative up to some fixed corrections. The details are in Proposition 4.30.

Yet already at this point we can recover something from the existence of the map a: uT → wT, namely an interpretation of the Alekseev-Torossian [AT] space of special derivations,

𝔰𝔡𝔢𝔯_{n} := {D ∈ 𝔱𝔡𝔢𝔯_{n}: D(∑_{i=1}^{n} x_{i}) = 0}.

Recall from Remark 3.14 that in general it is not possible to slide a strand under an arbitrary w-tangle. However, it is possible to slide strands freely under tangles in the image of a, and thus by reasoning similar to the reasoning in Remark 3.14, diagrams D in the image of α respect “tail-invariance”:

T  D +  + D =  ⇒ T  =  (9)

Let P^{u}(↑_{n}) denote the primitives of A^{u}(↑_{n}), that is, Jacobi diagrams that remain connected when the skeleton is removed. Remember that P^{w}(↑_{n}) stands for the primitives of A^{w}(↑_{n}). Equation (9) readily implies that the image of the composition

\xymatrix{P^{u}(↑_{n}) \ar[r]^{α} & P^{w}(↑_{n}) \ar[r]^{π} & 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n}}

is contained in 𝔞_{n} ⊕ 𝔰𝔡𝔢𝔯_{n}. Even better is true.

Theorem 3.28. The image of πα is precisely 𝔞_{n} ⊕ 𝔰𝔡𝔢𝔯_{n}.

This theorem was first proven by Drinfel’d (Lemma after Proposition 6.1 in [Dr3]), but the proof we give here is due to Levine [Lev]. Proof. Let 𝔩𝔦𝔢_{n}^{d} denote the degree d piece of 𝔩𝔦𝔢_{n}. Let V_{n} be the vector space with basis x_{1}, x_{2}, …, x_{n}. Note that

V_{n} ⊗ 𝔩𝔦𝔢_{n}^{d} ≅ ⨁_{i=1}^{n} 𝔩𝔦𝔢_{n}^{d} ≅ (𝔱𝔡𝔢𝔯_{n} ⊕𝔞_{n})^{d},

where 𝔱𝔡𝔢𝔯_{n} is graded by the number of tails of a tree, and 𝔞_{n} is contained in degree 1.

The bracket defines a map β: V_{n} ⊗ 𝔩𝔦𝔢_{n}^{d} → 𝔩𝔦𝔢_{n}^{d+1}: for a_{i} ∈ 𝔩𝔦𝔢_{n}^{d} where i = 1, …, n, the “tree” D = (a_{1}, a_{2}, …, a_{n}) ∈ (𝔱𝔡𝔢𝔯_{n} ⊕𝔞_{n})^{d} is mapped to

β(D) = ∑_{i=1}^{n}[x_{i}, a_{i}] = D (∑_{i=1}^{n} x_{i}) ,

where the first equality is by the definition of tensor product and the bracket, and the second is by the definition of the action of 𝔱𝔡𝔢𝔯_{n} on 𝔩𝔦𝔢_{n}.

Since 𝔞_{n} is contained in degree 1, by definition 𝔰𝔡𝔢𝔯_{n}^{d} = (ker β)^{d} for d ≥ 2. In degree 1, 𝔞_{n} is obviously in the kernel, hence (ker β)^{1} = 𝔞_{n} ⊕ 𝔰𝔡𝔢𝔯_{n}^{1}. So overall, ker β = 𝔞_{n} ⊕ 𝔰𝔡𝔢𝔯_{n}.

We want to study the image of the map P^{u}(↑^{n}) ⟶^{πα} 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n}. Under α, all connected Jacobi diagrams that are not trees or wheels go to zero, and under π so do all wheels. Furthermore, π maps trees that live on n strands to “floating” trees with univalent vertices coloured by the strand they used to end on. So for determining the image, we may replace P^{u}(↑^{n}) by the space T_{n} of connected unoriented “floating trees” (uni-trivalent graphs), the ends (univalent vertices) of which are coloured by the {x_{i}}_{i=1,..,n}. We denote the degree d piece of T_{n}, i.e., the space of trees with d + 1 ends, by T_{n}^{d}. Abusing notation, we shall denote the map induced by πα on T_{n} by α: T_{n} → 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n}. Since choosing a “head” determines the entire orientation of a tree by the two-in-one-out rule, α maps a tree in T_{n}^{d} to the sum of d + 1 ways of choosing one of the ends to be the “head”.

We want to show that ker β = im α. This is equivalent to saying that β̄ is injective, where β̄: V_{n} ⊗ 𝔩𝔦𝔢_{n} / im α → 𝔩𝔦𝔢_{n} is map induced by β on the quotient by im α.

x_{i} (0, ..., a_{i}, ..., 0) ↦^{β} ↦^{x_{i}}  [x_{i}, a_{i}]

The degree d piece of V_{n} ⊗ 𝔩𝔦𝔢_{n}, in the pictorial description, is generated by floating trees with d tails and one head, all coloured by x_{i}, i = 1, …, n. This is mapped to 𝔩𝔦𝔢_{n}^{d+1}, which is isomorphic to the space of floating trees with d + 1 tails and one head, where only the tails are coloured by the x_{i}. The map β acts as shown on the picture on the right.

τ  +

We show that β̄ is injective by exhibiting a map τ: 𝔩𝔦𝔢_{n}^{d+1} → V_{n} ⊗ 𝔩𝔦𝔢_{n}^{d} / im α so that τβ̄ = I. The map τ is defined as follows: given a tree with one head and d + 1 tails τ acts by deleting the head and the arc connecting it to the rest of the tree and summing over all ways of choosing a new head from one of the tails on the left half of the tree relative to the original placement of the head (see the picture on the right). As long as we show that τ is well-defined, it follows from the definition and the pictorial description of β that τβ̄ = I.

For well-definedness we need to check that the images of \overrightarrow{AS} and \overrightarrow{IHX} relations under τ are in the image of α. This we do in the picture below. In both cases it is enough to check the case when the “head” of the relation is the head of the tree itself, as otherwise an \overrightarrow{AS} or \overrightarrow{IHX} relation in the domain is mapped to an \overrightarrow{AS} or \overrightarrow{IHX} relation, thus zero, in the image.

\overrightarrow{AS} :  +  τ  +  +  +  +  ∈ im α

C^{B}  ^{C}^{B}  ^{C}^{B}^{A}  ^{A}  ^{A}  ^{C}^{B}^{A}  ^{C}^{B}  ^{C}^{B}^{A}  ^{A} −  +  + −  +  ^{C}^{B}  ^{C}^{B}^{A}  ^{A} −  = ↦^{τ} = −  ^{C}^{B}^{A}  ^{C}^{B}  ^{C}^{B}^{A}  ^{A} −  −  ∈ im α \overrightarrow{IHX} :

In the \overrightarrow{IHX} picture, in higher degrees A, B and C may denote an entire tree. In this case, the arrow at A (for example) means the sum of all head choices from the tree A. □

Comment 3.29. In view of the relation between the right half of Equation (9) and the special derivations 𝔰𝔡𝔢𝔯, it makes sense to call w-tangles that satisfy the condition in the left half of Equation (9) “special”. The a images of u-tangles are thus special. We do not know if the global version of Theorem 3.28 holds true. Namely, we do not know whether every special w-tangle is the a-image of a u-tangle.

3.4. The local topology of w-tangles. So far throughout this section we have presented w-tangles as a Reidemeister theory: a circuit algebra given by generators and relations. There is a topological intuition behind this definition: we can interpret the strands of a w-tangle diagram as embedded tubes in ℝ^{4} (with oriented “cores”: 1D curves that run along them), as shown in Figure 12. For each tube there exists a 3-dimensional “filling”, and each crossing represents a ribbon intersection between the filled tubes where the one corresponding to the under-strand intersects the filling of the over-strand. (For an explanation of ribbon singularities see [WKO1, Section 2.2.2].) In Figure 12 we use the drawing conventions of [CS]: we draw surfaces as if projected from ℝ^{4} to ℝ^{3}, and cut them open when they are “hidden” by something with a higher 4-th coordinate.

Figure 12. Strands correspond to tubes with cores, a virtual crossing corresponds to non-interacting tubes, while a crossing means that the tube corresponding to the under strand “goes through” the tube corresponding to the over strand.

Note that w-braids can also be thought of in terms of flying circles, with “time” being the fourth dimension; this is equivalent to the tube interpretation in the obvious way. In this language a crossing represents a circle (the under strand), flying through another (the over strand). This is described in detail in [WKO1, Section 2.2.1].

The assignment of tangled ribbon tubes in ℝ^{4} to w-tangles is well-defined (the Reidemeister and OC relations are satisfied), and after Satoh [Sa] we call it the tubing map and denote it by δ: {w-tangles} → {Ribbon tubes in ℝ^{4}}. It is natural to expect that δ is an isomorphism, and indeed it is a surjection. However, the injectivity of δ remains unproven even for long w-knots. Nonetheless, ribbon tubes in ℝ^{4} will serve as the topological motivation and local topological interpretation behind the circuit algebras presented in this paper.

Ribbon tubes in the image of δ are directed: where the direction comes from the direction of each tube as a strand of the tangle. In other words, each tube has a “core”^{11}: a distinguished line along the tube, which is oriented as a 1-dimensional manifold. An example is shown on the right.

Note that there are in fact four types (non-virtual) of crossings, given by whether the core of tube A intersects the filling of B or vice versa, and two possible directions in each case. In the flying ring interpretation, the orientation of the tube is the direction of the flow of time, and the four types of crossings represent: ring A flies through ring B from below or from above; and ring B flies through ring A from “below” or from “above” (cf. [WKO1, Exercise 2.7]). The two crossings not shown as generators are given by conjugating the classical crossing with virtual crossings, as in the bottom row of Figure 13.

ℝ^{3} : z = 0  z ℝ^{4}

We take the opportunity here to introduce another notation, to be called the “band notation”, which is more suggestive of the 4D topology than the strand notation we have been using so far. We represent a tube in ℝ^{4} by a picture of a directed band in ℝ^{3}. By “directed band” we mean that the band has a 1D direction (for example an orientation of one of the edges). To interpret the 3D picture of a band as an tube in ℝ^{4}, we add an extra coordinate. Let us refer to the ℝ^{3} coordinates as x, y and t, and to the extra coordinate as z. Think of ℝ^{3} as being embedded in ℝ^{4} as the hyperplane z = 0, and think of the band as being made of a thin double membrane. Push the membrane up and down in the z direction at each point as

11The core of Lord Voldemort’s wand was made of a phoenix feather.

Figure 13. Crossings and crossing signs in band notation.

far as the distance of that point from the boundary of the band, as shown on the right. This produces a tube embedded in ℝ^{4}. In band notation, the four types of crossings appear as in Figure 13, where underneath each crossing we indicate the corresponding strand picture.

3.5. Good properties and uniqueness of the homomorphic expansion. In much the same way as in the case of braids [WKO1, Section 2.5.1], Z has a number of good properties with respect to various tangle operations: it is group-like^{12}; it commutes with adding an inert strand (note that this is a circuit algebra operation, hence it doesn’t add anything beyond homomorphicity); and it commutes with deleting a strand and with strand orientation reversals. All but the last of these were explained in the context of braids and the explanations still hold. Orientation reversal S_{k}: wT → wT is the operation which reverses the orientation of the k-th component. In the image of Satoh’s tubing map this translates to reversing both the tube directions. The induced diagrammatic operation S_{k}: A^{w}(T) → A^{w}(S_{k}(T)), where T denotes the skeleton of a given w-tangle, acts by multiplying each arrow diagram by (−1) raised to the power the number of arrow endings (both heads and tails) on the k-th strand, as well as reversing the strand orientation. Saying that “Z commutes with S_{k}” means that the appropriate square commutes.

The following theorem asserts that a well-behaved homomorphic expansion of w-tangles is unique:

Theorem 3.30. The only homomorphic expansion satisfying the good properties described above is the Z defined in Section 3.1.

ρ =  +

Proof. We first prove the following claim: Assume, by contradiction, that Z′ is a different homomorphic expansion of w-tangles with the good properties described above. Let R′ = Z′(�) and R = Z(�), and denote by ρ the lowest degree homogeneous non-vanishing term of R′ −R. (Note that R′ determines Z′, so if Z′ ≠ Z, then R′ ≠ R.) Suppose ρ is of degree k. Then we claim that ρ = α_{1}w_{k}^{1}+α_{2}w_{k}^{2} is a linear combination of w_{k}^{1} and w_{k}^{2}, where w_{k}^{i} denotes a k-wheel living on strand i, as shown on the right.

Before proving the claim, note that it leads to a contradiction. Let d_{i} denote the operation “delete strand i”. Then up to degree k, we have d_{1}(R′) = α_{2}w_{k}^{1} and d_{2}(R′) = α_{1}w_{k}^{2}, but Z′ is compatible with strand deletions, so α_{1} = α_{2} = 0. Hence Z is unique, as stated.

On to the proof of the claim, note that Z′ being an expansion determines the degree 1 term of R′ (namely, the single arrow a^{12} from strand 1 to strand 2, with coefficient 1). So we can assume that k ≥ 2. Note also that since both R′ and R are group-like, ρ is primitive. Hence ρ is a linear combination of connected diagrams, namely trees and wheels.

12In practice this simply means that the value of the crossing is an exponential.

Both R and R′ satisfy the Reidemeister 3 relation:

R^{12}R^{13}R^{23} = R^{23}R^{13}R^{12}, R^{′12}R^{′13}R^{′23} = R^{′23}R^{′13}R^{′12}

where the superscripts denote the strands on which R is placed (compare with the proof of Theorem 3.10). We focus our attention on the degree k + 1 part of the equation for R′, and use that up to degree k + 1. We can write R′ = R + ρ + μ, where μ denotes the degree k + 1 homogeneous part of R′ − R. Thus, up to degree k + 1, we have

(R^{12}+ρ^{12}+μ^{12})(R^{13}+ρ^{13}+μ^{13})(R^{23}+ρ^{23}+μ^{23}) = (R^{23}+ρ^{23}+μ^{23})(R^{13}+ρ^{13}+μ^{13})(R^{12}+ρ^{12}+μ^{12}).

The homogeneous degree k + 1 part of this equation is a sum of some terms which contain ρ and some which don’t. The diligent reader can check that those which don’t involve ρ cancel on both sides, either due to the fact that R satisfies the Reidemeister 3 relation, or by simple degree counting. Rearranging all the terms which do involve ρ to the left side, we get the following equation, where a^{ij} denotes an arrow pointing from strand i to strand j:

[a^{12}, ρ^{13}] + [ρ^{12}, a^{13}] + [a^{12}, ρ^{23}] + [ρ^{12}, a^{23}] + [a^{13}, ρ^{23}] + [ρ^{13}, a^{23}] = 0.  (10)

The third and fifth terms sum to [a^{12} +a^{13}, ρ^{23}], which is zero due to the “head-invariance” of diagrams, as in Remark 3.14.

We treat the tree and wheel components of ρ separately. Let us first assume that ρ is a linear combination of trees. Recall that the space of trees on two strands is isomorphic to 𝔩𝔦𝔢_{2} ⊕ 𝔩𝔦𝔢_{2}, the first component given by trees whose head is on the first strand, and the second component by trees with their head on the second strand. Let ρ = ρ_{1} + ρ_{2}, where ρ_{i} is the projection to the i-th component for i = 1, 2.

Note that due to T C, we have [a^{12}, ρ_{2}^{13}] = [ρ_{2}^{12}, a^{13}] = [ρ_{1}^{12}, a^{23}] = 0. So Equation (10) reduces to

[a^{12}, ρ_{1}^{13}] + [ρ_{1}^{12}, a^{13}] + [ρ_{2}^{12}, a^{23}] + [ρ_{1}^{13}, a^{23}] + [ρ_{2}^{13}, a^{23}] = 0

The left side of this equation lives in ⨁_{i=1}^{3} 𝔩𝔦𝔢_{3}. Notice that only the first term lies in the second direct sum component, while the second, third and last terms live in the third one, and the fourth term lives in the first. This in particular means that the first term is itself zero. By \overrightarrow{ST U}, this implies

0 = [a^{12}, ρ_{1}^{13}] = −[ρ_{1}, x_{1}]_{2}^{13},

where [ρ_{1}, x_{1}]_{2}^{13} means the tree defined by the element [ρ_{1}, x_{1}] ∈ 𝔩𝔦𝔢_{2}, with its tails on strands 1 and 3, and head on strand 2. Hence, [ρ_{1}, x_{1}] = 0, so ρ_{1} is a multiple of x_{1}. The tree given by ρ_{1} = x_{1} is a degree 1 element, a possibility we have eliminated, so ρ_{1} = 0.

Equation (10) is now reduced to

[ρ_{2}^{12}, a^{23}] + [ρ_{2}^{13}, a^{23}] = 0.

Both terms are words in 𝔩𝔦𝔢_{3}, but notice that the first term does not involve the letter x_{3}. This means that if the second term involves x_{3} at all, i.e., if ρ_{2} has tails on the second strand, then both terms have to be zero individually. Assuming this and looking at the first term, ρ_{2}^{12} is a Lie word in x_{1} and x_{2}, which does involve x_{2} by assumption. We have [ρ_{2}^{12}, a^{23}] = [x_{2}, ρ_{2}^{12}] = 0, which implies ρ_{2}^{12} is a multiple of x_{2}, in other words, ρ is a single arrow on the second strand. This is ruled out by the assumption that k ≥ 2.

On the other hand if the second term does not involve x_{3} at all, then ρ_{2} has no tails on the second strand, hence it is of degree 1, but again k ≥ 2. We have proven that the “tree part” of ρ is zero.

So ρ is a linear combination of wheels. Wheels have only tails, so the first, second and fourth terms of (10) are zero due to the tails commute relation. What remains is [ρ^{13}, a^{23}] = 0. We assert that this is true if and only if each linear component of ρ has all of its tails on one strand.

To prove this, recall each wheel of ρ^{13} represents a cyclic word in letters x_{1} and x_{3}. The map r: ρ^{13} ↦ [ρ^{13}, a^{23}] is a map 𝔱𝔯_{2} → 𝔱𝔯_{3}, which sends each cyclic word in letters x_{1} and x_{3} to the sum of all ways of substituting [x_{2}, x_{3}] for one of the x_{3}’s in the word. Note that if we expand the commutators, then all terms that have x_{2} between two x_{3}’s cancel. Hence all remaining terms will be cyclic words in x_{1} and x_{3} with a single occurrence of x_{2} in between an x_{1} and an x_{3}.

We construct an almost-inverse r′ to r: for a cyclic word w in 𝔱𝔯_{3} with one occurrence of x_{2}, let r′ be the map that deletes x_{2} from w and maps it to the resulting word in 𝔱𝔯_{2} if x_{2} is followed by x_{3} in w, and maps it to 0 otherwise. On the rest of 𝔱𝔯_{3} the map r′ may be defined to be 0.

The composition r′r takes a cyclic word in x_{1} and x_{3} to itself multiplied by the number of times a letter x_{3} follows a letter x_{1} in it. The kernel of this map can consist only of cyclic words that do not contain the sub-word x_{3}x_{1}, namely, these are the words of the form x_{3}^{k} or x_{1}^{k}. Such words are indeed in the kernel of r, so these make up exactly the kernel of r. This is exactly what needed to be proven: all wheels in ρ have all their tails on one strand.

This concludes the proof of the claim, and the proof of the theorem. □

4. w-Tangled Foams

Section Summary. In this section we add “foam vertices” to w-tangles (and a few lesser things as well) and ask the same questions we asked before; primarily, “is there a homomorphic expansion?”. As we shall see, in the current context this question is equivalent to the Alekseev-Torossian [AT] version of the Kashiwara-Vergne [KV] problem and explains the relationship between these topics and Drinfel’d’s theory of associators. This section was corrected and improved post-publication: see Section 1.3 and [WKO2C].

4.1. The Circuit Algebra of w-Tangled Foams. In the same manner as we did for tangles, we present the circuit algebra of w-tangled foams via its Reidemeister-style diagrammatic description accompanied by a local topological interpretation. To give a finite presentation for a circuit algebra with auxiliary (additional) operations, we use the notation

CA ⟨ Circuit algebra generators | Circuit algebra relations | Auxiliary operations ⟩ .

Definition 4.1. Let wTF denote the circuit algebra given by the following generators, relations and auxiliary operations:

wTF = CA⟨  , ^{w}, , ,  ,  | R1^{s}, R2, R3, R4, OC, CP, FR, W^{2}, CW, TV | S_{e}, u_{e}, d_{e} ⟩ .

The generators consist of crossings, caps, wens, and foam vertices. Note that the foam vertices, where three strands meet, also come in all possible combinations of strand directions. Some additional examples are shown on the right. These generators are related by the orientation switch operation S_{e}, whose topological interpretation is explained in Section 4.2.3.

The relations R1^{s}, R2, R3 and OC are as in Section 3. The other relations are shown and explained in the context of their local topological meaning in Section 4.2.2.

An edge of a w-tangled foam is a line between two vertices, tangle ends (boundary points), or caps; edges may go over and under multiple crossings. The auxiliary operations of wTF are edge orientation switches S_{e}, edge unzips u_{e}, and deletions d_{e} of long strands which end in two tangle ends. These are described, along with their topological interpretations, in Section 4.2.3.

The circuit algebra wTF is skeleton-graded where the circuit algebra of skeleta S is a version of the skeleton algebra S introduced in Section 2.4, but with vertices, caps and wens included:

S = CA⟨ , , , ^{w}  | W^{2}, CW, TV ⟩ .

Denote by σ : wTF → S the skeleton map, given by σ(�) = σ(�) = �, and where all other generators are mapped to themselves.

4.2. The local topology of w-tangled foams. In this section we present the local topological meaning of wTF generators, present the relations and show that they represent local isotopies for a space of ribbon-embedded tubes in ℝ^{4} with caps, wens (that is, open Klein bottles), and foam vertices. We interpret the auxiliary operations as topological operations on this space.

Comment 4.2. We conjecture that the generators and relations of wTF provide a Reidemeister theory for this topological interpretation of w-tangled foams. However, there is no complete Reidemeister theorem even for w-knots (see [WKO1, Section 3]). For any rigorous purposes below, wTF is studied as a circuit algebra given by generators and relations, with topology serving only as intuition.

4.2.1. The generators of wTF. There is topological meaning to each of the generators of wTF: via a generalization of the Satoh tubing map δ of Section 3 they each stand for local features of framed knotted ribbon tubes in ℝ^{4}.

The map δ treats the strands in the same way as in Section 3.4. The crossings are also as explained in Section 3.4: the under-strand denotes the small circle flying through a larger one, or, equivalently, a “thin” tube braided through a thicker one. Recall that for tangles there are four kinds of crossings (left or right circle flying through from below or from above the other). Two of these are the generators shown, and the other two are obtained from the generators by adding virtual crossings (see Figures 12 and 13).

A bulleted end denotes a cap on the tube, or a flying circle that shrinks to a point, as in the figure on the right.

The w marking on a strand indicates a wen. A wen is a Klein bottle cut apart (see [WKO1, Section 2.5.4]). In the flying circle perspective, a wen represents a circle which flips over and at the same time changes its orientation; in the band persepctive, a twisted band.

w  =

The final generators denote singular foam vertices. As the notation suggests, a vertex can be thought of as a crossing with either the bottom or the top half tubes identified. To make this precise using the flying circles interpretation, the vertex represents the movie shown on the left: the circle corresponding to the right strand approaches the ring represented by the left strand from below, flies inside it, and then the two rings fuse (as opposed to a crossing where the ring coming from the right would continue to fly out to above and to the left of the other one). The second vertex is the movie where a ring splits radially into a smaller and a larger ring, and the small one flies out to the right and below the big one. The edge corresponding to two rings identified (i.e., the top edge of and the bottom edge of ) is called the stem.

Vertices are rigid: the three tubes meeting at a vertex play different roles. Combinatorially, this means that the three edges meeting at a vertex are labelled (stem, inner and outer), and carry a cyclic orientation. In practice we use asymmetric pictures to avoid the clutter of labels.

As with crossings, we obtain the vertices with opposite fly-in directions by composing the generating vertices with virtual crossings, as shown in Figure 14. In the figure the band notation for vertices is used the same way as it is for crossings: the fully coloured band stands for the thin (inner) ring.

Figure 14. Vertex types in wTF^{o}.

4.2.2. The relations of wTF. Next, we discuss the relations of wTF and show that they represent local isotopies of w-foams. The usual R1^{s}, R2, R3, and OC relations of Figure 3 continue to apply.

The Reidemeister 4 (R4) relations assert that a strand can be moved under or over a vertex, as shown below. The ambiguously drawn vertices in the figure denote a vertex of any sign with any strand directions (as in Section 4.2.1). The local isotopies can be read from the band pictures in the bottom row.

R4 :

Recall that topologically, a cap represents a capped tube or equivalently, flying circle shrinking to a point. Hence, a cap on the thin (or under) strand can be “pulled out” from a crossing, but the same is not true for a cap on the thick (or over) strand, as shown below. We denote this relation by CP, for Cap Pull-out. This is the case for any strands directions.

.. ,, CP :  yet yet

The FR, W^{2}, CW and TV relations describe the behaviour of the wens, and together we refer to them as the wen relations.

The interaction of a wen and a crossing is described by the following Flip Relations (FR):

A B  A B  A B  A B = w  w  w  w

To explain this in the flying circle interpretation, recall that a wen represents a circle that flips over. It does not matter whether ring B flips first and then flies through ring A or vice versa. However, the movies in which ring A first flips and then ring B flies through it, or B flies through A first and then A flips differ in the fly-through direction of B through A, hence the virtual crossings.

A double flip is isotopic to no flip, in other words two consecutive wens are isotopic to no wen. We denote this relation by W^{2}.

w

A cap can slide through a wen, hence a capped wen disappears, as shown on the right, to be denoted CW.

⟷ ⟷  ⟷ w w w

The last wen relation describes the interaction of wens and vertices, as illustrated on the left. In the band notation the non-filled band represents the larger circle, and the band the inner/smaller circle, as usual. Conjugating a vertex by three wens switches the top and bottom bands, as shown in the figure on the left. Alternatively in the flying circle interpretation, if both rings flip, then merge, and then the

merged ring flips again, this is isotopic to no flips, except the fly-in direction (from below or from above) has changed. We denote the diagrammatic relation arising from this isotopy – shown in the bottom right – by TV, for Twisted Vertex.

4.2.3. The auxiliary operations of wTF. The circuit algebra wTF is equipped with several auxiliary operations.

The first of these is the familiar orientation switch: given an edge e of a w-foam, S_{e} switched the direction of the edge e.

e  u_{e}

The most interesting operation on w-foams is the edge unzip u_{e}, which doubles the edge e using the blackboard framing, then attaches the ends of the doubled edge to the connecting ones, as shown on the right. Unzip is only defined when the directions of the edges involved match, as shown on the left. We restrict unzip to edges that are the stem of each of the two vertices they connect, and whose two vertices have their inner and outer edges aligned, as shown (in other words, edges which connect two different generating vertices).

Comment 4.3. In order to understand the local topological meaning of the unzip operation, we need to discuss framings in more depth. Recall that framings were mentioned in Section 3.4, but have not played a significant role so far, except to explain the lack of a Reidemeister 1 relation.

In the local topological interpretation of wTF, edges represent ribbon-knotted tubes with foam vertices, which are also equipped with a framing, arising from the blackboard framing of the strand diagrams via Satoh’s tubing map. Topologically, unzip is the operation of doubling a tube by “pushing it off itself slightly” in the framing direction, as shown in Figure 15.

Recall that ribbon knotted tubes have a “filling”, with only “ribbon” self-intersections [WKO1, Section 2.2.2]. When we double a tube, we want this ribbon property to be preserved. This is equivalent to saying that the circle obtained by pushing off any given girth of the tube in the framing direction is not linked with the original tube, which is indeed the case.

Figure 15. Unzipping a tube, in band notation with orientations and framing marked.

Framings arising from the blackboard framing of strand diagrams via Satoh’s tubing map always match at the vertices, with the normal vectors pointing either directly towards or away from the center of the singular ring. Note that the directions of the three tubes may or may not match. An example of a vertex with the orientations and framings shown is on the right. Note that the framings on the two sides of each band are always mirror images of each other.

When a tube is unzipped, at each of the vertices at the two ends of the doubled tube there are two tubes to be attached to the doubled tube. At each end, the normal vectors pointed either directly towards or away from the center, so there is an “inside” and an “outside” boundary circle. The two tubes to be attached also come as an “inside” and an “outside” one, which defines which one to attach to which. An example is shown in Figure 15.

A related operation, disk unzip, is unzip done on a capped edge, pushing the edge off in the direction of the blackboard framing, as before. An example in the line and band notations (with the framing suppressed) is shown below.

e  u_{e}  e  u_{e}  =

Note that edges which contain wens may be unzipped by first relocating the wens to other edges or removing them, using the wen relations.

The edge deletion (denoted d_{e}) operation is restricted to “long linear” edges, meaning edges that do not end in a vertex or cap.

4.3. The Associated Graded Structure. Mirroring the previous section, we describe the associated graded structure A^{sw} of wTF and its “full version” A^{w} as circuit algebras on certain generators modulo a number of relations. From now on we will write A^{(s)w} to mean “A^{w} and/or A^{sw}”.

A^{(s)w} = CA ⟨  , , , ,^{w}  | \overrightarrow{4T}, TC, VI, CP, W^{2}, TW, CW, FR, (RI for A^{sw}) | S_{e}, u_{e}, d_{e} ⟩ .

In other words, A^{(s)w} is the circuit algebra of arrow diagrams on trivalent (foam) skeleta with caps and wens. That is, the skeleta are elements of S as in Section 4.1. With the exception of the first generator (the arrow), all generators are skeleton features (of degree 0). The arrow is of degree 1. As for the generating vertices, the same remark applies as in Definition 4.21, that is, vertices come in all possible combinations of edge directions.

4.3.1. The relations of A^{(s)w}. In addition to the usual \overrightarrow{4T} and TC relations (see Figure 5), as well as RI in the case of A^{sw} = A^{w}/RI, arrow diagrams in A^{(s)w} satisfy the following additional relations:

Vertex invariance, denoted by VI, are relations which arise from the same principle as the classical \overrightarrow{4T} relation, but with a vertex in place of a crossing:

±  ±  ±  ± ±  ±  = 0,  and  = 0.^{VI}  ^{VI}

The other end of the arrow is in the same place throughout the relation, somewhere outside the picture shown. The signs are positive whenever the edge on which the arrow ends is directed towards the vertex, and negative when directed away. The ambiguously drawn vertex means any kind of vertex, with edges oriented in any direction, as long as it is the same one throughout the relation.

= 0

The CP relation (a cap can be pulled out from under a edge but not from over, Section 4.2.2) implies that arrow heads vanish next to a cap, as shown on the right. We denote this relation also by CP. (Note that an arrow tail near a cap may not vanish.)

The W^{2}, TV, and CW relations are skeleton relations introduced in Section 4.2.2 which describe the interactions of wens with each other, vertices and caps, and they continue to apply to the skeleta of arrow diagrams.

The Flip Relations FR imply that wens “commute” with arrow heads, but “anti-commute” with arrow tails. We denote the associated graded Flip Relations also by FR.

FR:  w  = w  but w  = − w .

As in Definition 3.12, we define a “w-Jacobi diagram” (or just “arrow diagram”) on a foam skeleton by allowing trivalent arrow vertices. Denote the circuit algebra of formal linear combinations of arrow diagrams, modulo the relations of A^{(s)w} and the \overrightarrow{ST U} relations of Figure 6, by A^{(s)wt}. We have the following bracket-rise theorem:

Theorem 4.4. The natural inclusion of diagrams induces a circuit algebra isomorphism A^{(s)w} ≅ A^{(s)wt}. Furthermore, the \overrightarrow{AS} and \overrightarrow{IHX} relations of Figure 7 hold in A^{(s)wt}.

Proof. Same as the proof of Theorem 3.13. □

As in Section 3.1, the primitive elements of A^{(s)w} are connected diagrams (that is, connected with the skeleton removed), which are linearly generated by trees and wheels. Before moving on to the auxiliary operations of A^{(s)w}, we make two useful observations:

Lemma 4.5. A^{w}( ), the part of A^{w} with skeleton , is isomorphic as a vector space to the completed polynomial algebra freely generated by wheels w_{k} with k ≥ 1. Likewise A^{sw}( ), except here k ≥ 2.

Proof. Any arrow diagram with an arrow head at its top is zero by the Cap Pull-out (CP) relation. If D is an arrow diagram that has a head somewhere on the skeleton but not at the top, then one can use repeated \overrightarrow{ST U} relations to commute the head to the top at the cost of diagrams with one fewer skeleton head.

Iterating this procedure, we can get rid of all arrow heads, and hence write D as a linear combination of diagrams having no heads on the skeleton. All connected components of such diagrams are wheels.

To prove that there are no relations between wheels in A^{(s)w}( ), let S_{L}: A^{(s)w}(↑_{1}) → A^{(s)w}(↑_{1}) (resp. S_{R}) be the map that sends an arrow diagram to the sum of all ways of dropping one left (resp. right) arrow (on a vertical edge, left means down and right means up). Define

F := ∑_{k=0}^{∞} \frac{(−1)^{k}}{k!} D_{R}^{k}(S_{L} + S_{R})^{k},

where D_{R} is a short right arrow. We leave it as an exercise for the reader to check that F is a bi-algebra homomorphism that kills diagrams with an arrow head at the top (i.e., CP is in the kernel of F), and F is injective on wheels. This concludes the proof. □

Lemma 4.6. A^{(s)w}( ) = A^{(s)w}(↑_{2}), where A^{(s)w}( ) stands for the space of arrow diagrams whose skeleton is a vertex of any type, with any orientation of the edges, and A^{(s)w}(↑_{2}) denotes the space of arrow diagrams on two strands.

Proof. Use the vertex invariance (VI) relation to push all arrow heads and tails from the “trunk” of the vertex to the other two edges. □

4.3.2. The auxiliary operations of A^{(s)w}. Recall from Section 3.5 that the associated graded orientation switch operation S_{e}: A^{(s)w}(s) → A^{(s)w}(S_{e}(s)) acts by reversing the direction of the skeleton edge e, and multiplying each arrow diagram by (−1) raised to the number of arrow endings on e (counting both heads and tails).

e  u_{e} +

The arrow diagram operations induced by unzip and disc unzip u_{e}: A^{(s)w}(s) → A^{(s)w}(u_{e}(s)) are both denoted u_{e}, and interpreted appropriately according to whether the edge e is capped. They both map each arrow ending (head or tail) on e to a sum of two arrows, one ending on each of the new edges, as shown on the right. In other words, if in a primitive arrow diagram D there are k arrow ends on e, then u_{e}(D) is a sum of 2^{k} primitive arrow diagrams.

The operation induced by deleting the long linear strand e is the map d_{e}: A^{(s)w}(s) → A^{(s)w}(d_{e}(s)) which kills arrow diagrams with any arrow ending (head or tail) on e, and leaves all else unchanged, except with e removed.

4.4. The homomorphic expansion. If a homomorphic expansion for wTF exists, it is determined by the values of the generators, as wTF has a finite presetantion. We are interested in particular in group-like homomorphic expansions, where the values of the generators are exponentials of (infinite series of) primitive arrow diagrams. For more detail see [WKO1, Section 2.5.1.2].

We will see that the value of the vertex V ∈ A^{sw}( ) ≅ A^{sw}(↑_{2}) plays a particularly important role. It will also become clear that the short arrows of V can be ignored: here a short arrow means an arrow on a single edge of the vertex, whose head and tail are adjacent on the skeleton. The reason is that, given a homomorphic expansion Z with Z( ) = V = e^{v}, and a is a short arrow on the vertex, then changing V to V′ = e^{v+a} defines another homomorphic expansion Z′. We explain this in more detail later; for now we make the following definition:

Definition 4.7. A homomorphic expansion is v-small if Z( ) = e^{v} where v is a (possibly infinite) linear combination of primitive arrow diagrams which does not include short arrows in degree one.

Given a homomorphic expansion Z : wTF → A^{sw}, denote by W the Z-value of the wen. We are now able to state one of the main theorems of this paper:

Theorem 4.8. Group-like^{13} homomorphic expansions Z : wTF → A^{sw} exist, and these which are v-small and satisfy W = 1 are in one-to one correspondence with solutions to the Kashiwara-Vergne equations (defined in Section 4.5) with even Duflo function. □

Our goal is to explain and prove this theorem. To begin, observe that finding a homomorphic expansion Z : wTF → A^{sw} is equivalent to finding values for the generators of wTF in A^{sw}, so that these values satisfy the equations which arise from the relations in wTF and the homomorphicity with respect to the auxiliary operations. In this subsection we derive these equations; in Section 4.5 we show that they are equivalent to the Alekseev-Torossian version of the Kashiwara-Vergne equations [AT] with even Duflo function. In [AET] Alekseev Enriquez and Torossian construct explicit solutions to these equations using associators. In [WKO3] we will interpret and independently prove this result in the context of homomorphic expansions for w-tangled foams.

First we set notation for the images of the most important generators. Assume that Z is a homomorphic expansion. Let R := Z(�) ∈ A^{sw}(↑_{2}). Let C := Z( ) ∈ A^{sw}( ). By Lemma 4.5, we know that C is made up of wheels only. Let W ∈ A^{sw}(↑) denote the Z-value of the wen, and we adopt the convention that W is always placed on the skeleton edge after the wen. Finally, let V = V^{+} := Z( ) ∈ A^{sw}( ) ≅ A^{sw}(↑_{2}), and V^{−} := Z( ) ∈ A^{sw}( ) ≅ A^{sw}(↑_{2}).

We first address the value of the wen. Recall that the FR relation in A^{sw} states that skeleton wens commute with arrow heads and anti-commute with arrow tails. For a primitive arrow diagram D we denote

\overline{D} := (−1)^{#\{arrow tails in D\}}D.

Then we have that wD = \overline{D}w, where w denotes a skeleton wen and D ∈ A^{sw}(↑). The same equality holds in A^{sw}(↑_{n}) if all strands of D are commuted with wens.

Lemma 4.9. Under any group-like homomorphic expansion Z the value of the wen can be expressed as W = exp(∑_{k=1}^{∞} c_{2k+1}w_{2k+1}), where c_{2k+1} are constants, and w_{2k+1} are odd wheels, and any such W satisies the equations induced by the W^{2} relation and homomorphicity with respect to S_{e}.

Proof. From the W^{2} relation we obtain that \overline{W} W = 1, see Figure 16. Since Z is group-like, we have W = e^{ω} for some primitive ω. Since ω is a primitive element of A^{sw}(↑), by the description of primitive arrow diagrams in Section 3.2 it can be written a sum of wheels

13The formal definition of the group-like property is along the lines of [WKO1, Section 2.5.1.2]. In practice, it means that the Z-values of the vertices, crossings, and cap (denoted V, R and C below) are exponentials of linear combinations of connected diagrams.

= w w = 1  W W  W \overline{W}

Figure 16. The implication of the W^{2} relation.

in degrees 2 and above, with possibly a multiple of a single arrow in degree 1. (Higher degree trees on a single strand reduce to wheels by the AS and STU relations.) Write ω = p_{1}a + ∑_{k=1}^{∞} p_{2k}w_{2k} + ∑_{l=1}^{∞} p_{2l+1}w_{2l+1}, where a denotes the degree 1 arrow, w_{i} are iwheels, and p_{i} are constants.

Then \overline{W} = e\overline{ω}, and \overline{ω} = −p_{1}a + ∑_{k=1}^{∞} p_{2k}w_{2k} − ∑_{l=1}^{∞} p_{2l+1}w_{2l+1}. Thus, \overline{W} = W^{−1} means that p_{2k} = 0 for all k ≥ 1, in other words, W is contained in odd degrees only.

From the homomorphicity of Z with respect to orientation switches we further see that S(W) = \overline{W′}, where S(W) denotes the orientation switch of W. Combining this with the previous result, we have S(W) = \overline{W}, which further implies that p_{1} = 0, completing the proof. □

Recall that by convention we number strands at the bottom of each diagram from left to right, and for an arrow diagram D ∈ A^{sw}(↑_{k}), D^{i_{1}i_{2}...i_{k}} means “D placed on strands i_{1}, …, i_{k}. For instance, R^{23} means “R placed on strands 2 and 3”. In this section we also need to use co-simplicial notation, for example R^{(23)1} means “R with its first strand doubled (unzipped), then placed on strands 2, 3 and 1”.

We recall the following result Sections 3.1 and 3.5:

Lemma 4.10. For any homomorphic expansion Z, the values of the crossings are as follows: R = Z(�) = e^{a} where a denotes a single arrow from the over to the under strand, Z(�) = (R^{−1})^{21} = e^{−a^{21}}, where again a^{21} points from the over strand to the under strand. These values satisfy the equations induced by R1^{s}, R2, R3 and OC in A^{sw}. □

One of the important restrictions on the value V arises from the R4 relations:

Lemma 4.11. The R4 relations induce the following single equation on V and R:

V^{12}R^{(12)3} = R^{23}R^{13}V^{12}.  (R4)

Proof. The Reidemeister 4 relation with a strand over a vertex induces an equation that is automatically satisfied, as follows:

Z  =  V V R  R  = + V  R R V R R  VI

In other words, the over strand R4 relation induces the equation

V^{12}R^{3(12)} = R^{32}R^{31}V^{12}.

However, observe that by the “head-invariance” property of arrow diagrams (Remark 3.14) V^{12} and R^{3(12)} commute on the left hand side. Hence the left hand side equals R^{3(12)}V^{12} = R^{32}R^{31}V^{12}. Also, R^{3(12)} = e^{a^{31}+a^{32}} = e^{a^{32}}e^{a^{31}} = R^{32}R^{31}, where the second step is an application of the TC relation (a^{31} and a^{32} commute). Therefore, this equation is true regardless of the choice of V.

We have no such luck with the second Reidemeister 4 relation, which, in the same manner as above, translates to the (R4) equation V^{12}R^{(12)3} = R^{23}R^{13}V^{12}. There is no “tail invariance” of arrow diagrams, so V and R do not commute on the left hand side; also, heads do not commute and so R^{(12)3} ≠ R^{23}R^{13}. Thus, this equation places a genuine restriction on the choice of V. □

Lemma 4.12. The equation induced by the CP relation is automatically satisfied for any choice of C.

Proof. The Cap Pull-out (CP) relation translates to the equation R^{12}C^{2} = C^{2}. By head invariance, R^{12}C^{2} = C^{2}R^{12}. Now R^{12} is just below the cap on strand 2, and thus by the CP relation in A^{sw}, every term of R^{12} with an arrow head at the top of strand 2 is zero. Hence, the only surviving term of R^{12} is 1 (the empty diagram), which makes the equation true. □

Lemma 4.13. If V = Z( ), and V^{−} = Z( ), then V^{−} = V^{−1}.

Proof. This is an immediate consequence of the homomorphicity of Z with respect to the unzip operation. □

For the value of the cap denote C = e^{c}, where c = ∑_{j=1}^{∞} r_{j}w_{j}, with r_{j} constants and w_{j} the j-wheel. The value of the cap is the product of even and odd parts, that is, C = C_{eve}C_{odd}, where C_{eve} = e^{c_{eve}} with c_{eve} = ∑_{k=1}^{∞} r_{2k}w_{2k}, and C_{odd} = e^{c_{odd}} with c_{odd} = ∑_{l=1}^{∞} r_{2k}w_{2k}

Lemma 4.14. The equation induced by the CW relation is c_{odd} = −\frac{1}{2}ω, or equivalently C_{odd} = W^{−1/2}.

Proof. Applying Z to each side of the CW relation, we obtain \overline{W} \overline{C} = C in A^{sw}(�). Substituting the formulas for W and C, the statement follows. □

Possibly the most interesting equation is the one induced by the twisted vertex relation. For this we introduce one additional piece of notation. Given D ∈ A^{sw}(↑_{n}), denote D^{∗} := S_{1}S_{2} ⋯ S_{n}(\overline{D}), and call this the adjoint of D. In other words, the operation ∗ : A^{sw}(↑_{n}) → A^{sw}(↑_{n}) reverses the edge directions and multiplies an arrow diagram D by (−1)^{#{arrow heads on the skeleton}} .

Lemma 4.15. The TV relation is induces the “Wen-Unitarity” equation

(W^{−1})^{(12)}V^{∗}W^{1}W^{2}V = 1.  (WU)

Proof. We apply Z to each side of the TV relation, as shown in Figure 17. On the right hand side of the relation is a vertex with the edge orientations reversed, upside down and the edges numbered (2, 1) as the vertex follows a virtual crossing. Therefore, the value of this vertex is S_{1}S_{2}(V^{−1}) by Lemma 4.13.

The top wen value W on the far left side can be “pulled down” to the bottom two edges using the VI relation. Therefore, we obtain the following equation in A^{sw}(↑_{2}):

\overline{W}^{1}\overline{W}^{2}\overline{V} W^{(12)} = S_{1}S_{2}(V^{−1})

=  S_{1}S_{2}(V^{−1}) w w w  2 1 1  2 2 1 V W W w w w  2 1 = \overline{V} \overline{W} \overline{W} \overline{W} 1  2  Z T V Z

Figure 17. Applying Z to each side of the TV relation.

Applying S_{1}S_{2} to both sides, and using from Lemma 4.9 that W̄ = W^{−1} = S(W), we obtain the equation (WU). □

Lemma 4.16. Homomorphicity of Z with respect to the disc unzip operation is equavialent to the Cap Equation:

V^{12}C^{(12)} = C^{1}C^{2}  in A^{sw}(_{2})  (C)

u u ∘ Z^{w}  Z^{w} V C^{(12)}  C^{1}C^{2}

Proof. We need to apply Z and the cap unzip u in either order to the w-foam shown in the figure on the right. On the left hand side, the value of the cap is unzipped and gives C^{(12)}. Note that (C) is an equation in A^{sw}(�_{2}). □

To summarize, we have proven the following theorem:

Theorem 4.17. Z : wTF → A^{sw} is a group-like homomorphic expansion if and only if the values of the key generators R = e^{a}, V, W and C are group-like, and satisfy the equations (R4) V^{12}R^{(12)3} = R^{23}R^{13}V^{12} in A^{sw}(↑_{3}), (WU) (W^{−1})^{(12)}V^{∗}W^{1}W^{2}V = 1 in A^{sw}(↑_{2}),

(C) V^{12}C^{(12)} = C^{1}C^{2} in A^{sw}(�_{2}), (CW) C_{odd} = W^{−1/2} in A^{sw}(�_{1}), or equivalently, as power series in odd wheels.

4.5. The equivalence with the Alekseev-Torossian equations. First let us recall Alekseev and Torossian’s formulation of the generalized Kashiwara-Vergne problem (see [AT, Section 5.3]):

Generalized KV problem: Find an element F ∈ TAut_{2} with the properties

F (x + y) = log(e^{x}e^{y}), and j(F) ∈ im(δ̃).  (11)

Here δ̃: 𝔱𝔯_{1} → 𝔱𝔯_{2} is defined by (δ̃a)(x, y) = a(x)+a(y)−a(log(e^{x}e^{y})), where 𝔱𝔯_{2} is generated by cyclic words in the letters x and y. (See [AT], Equation (8)). Note that an element of 𝔱𝔯_{1} is a power series in one variable with no constant term, called the Duflo function. In other words, the second condition says that there exists a ∈ 𝔱𝔯_{1} such that jF = a(x) + a(y) − a(log(e^{x}e^{y})).

Proof of Theorem 4.8. We need to translate the equations of Theorem 4.17 to equations in the Alekseev-Torossian spaces, using the identifications of Proposition 3.19 and the identification of wheels with cyclic words. Note the condition in Theorem 4.8 that W = 1. With this simplification the (CW) equation simply asserts that the value C is an even power series in wheels. The (WU) equation simplifies to the following, which we call the Unitarity of the vertex:

V^{∗}V = 1.  (U)

Recall from Section 3.2 that the map u: 𝔱𝔡𝔢𝔯_{2} → A^{sw}(↑_{2}) plants the head of a tree above all of its tails. Suppose that the values V and C satisfy the simplified equations of Theorem 4.17 with W = 1. Write V = e^{b}e^{uD}, where b ∈ tr_{2}^{s}, D ∈ 𝔱𝔡𝔢𝔯_{2} ⊕𝔞_{2}, and where V can be written in this form without loss of generality because wheels can always be commuted to the bottom of a diagram (at the possible cost of more wheels). Furthermore, V is group-like and hence it can be written in exponential form. Similarly, write C = e^{c} with c ∈ 𝔱𝔯_{1}^{s}.

Note that u(𝔞_{2}) is central in A^{sw}(↑_{2}) and that replacing a solution (V, C) by (e^{u(a)}V, C) for any a ∈ 𝔞_{2} does not interfere with any of the equations (R4), (U) or (C). Hence we may assume that D does not contain any single arrows, that is, Z is v-small and D ∈ 𝔱𝔡𝔢𝔯_{2}. Also, a solution (V, C) in A^{sw} can be lifted to a solution in A^{w} by simply setting the degree one terms of b and c to be zero. It is easy to check that this b ∈ 𝔱𝔯_{2} and c ∈ 𝔱𝔯_{1} along with D still satisfy the equations. (In fact, in A^{w} (U) and (C) respectively imply that b is zero in degree 1, and c is already assumed to be even.) In light of this we declare that b ∈ 𝔱𝔯_{2} and c ∈ 𝔱𝔯_{1}.

The hard Reidemeister 4 equation (R4) reads V^{12}R^{(12)3} = R^{23}R^{13}V^{12}. Denote the arrow from strand 1 to strand 3 by x, and the arrow from strand 2 to strand 3 by y. Substituting the known value for R and rearranging, we get

e^{b}e^{uD}e^{x+y}e^{−uD}e^{−b} = e^{y}e^{x}.

Equivalently, e^{uD}e^{x+y}e^{−uD} = e^{−b}e^{y}e^{x}e^{b}. Now on the right side there are only tails on the first two strands, hence e^{b} commutes with e^{y}e^{x}, so e^{−b}e^{b} cancels. Taking logarithm of both sides we obtain e^{uD}(x + y)e^{−uD} = log e^{y}e^{x}. Now for notational alignment with [AT] we switch strands 1 and 2, which exchanges x and y so we obtain:

e^{uD^{21}}(x + y)e^{−uD^{21}} = log e^{x}e^{y}.  (12)

The unitarity of V (Equation (U)) translates to 1 = e^{b}e^{uD}(e^{b}e^{uD})^{∗}, where ∗ denotes the adjoint map (Definition 3.26). Note that the adjoint switches the order of a product and acts trivially on wheels. Also, e^{uD}(e^{uD})^{∗} = J(e^{D}) = e^{j(e^{D})}, by Proposition 3.27. So we have 1 = e^{b}e^{j(e^{D})}e^{b}. Multiplying by e^{−b} on the right and by e^{b} on the left, we get 1 = e^{2b}e^{j(e^{D})}, and again by switching strand 1 and 2 we arrive at

1 = e^{2b^{21}}e^{j(e^{D^{21}})}.  (13)

As for the cap equation, if C^{1} = e^{c(x)} and C^{2} = e^{c(y)}, then C^{12} = e^{c(x+y)}. Note that wheels on different strands commute, hence e^{c(x)}e^{c(y)} = e^{c(x)+c(y)}, so the cap equation reads

e^{b}e^{uD}e^{c(x+y)} = e^{c(x)+c(y)}.

As this equation lives in the space of arrow diagrams on two capped strands, it remains unchanged if we multiply the left side on the right by e^{−uD}: uD has its head at the top, so it is 0 by the Cap relation, hence e^{uD} = 1 near the cap. Hence,

e^{b}e^{uD}e^{c(x+y)}e^{−uD} = e^{c(x)+c(y)}.

σ

On the right side of the equation above e^{uD}e^{c(x+y)}e^{−uD} reminds us of Equation (12), however we cannot use (12) directly as we are working in a different space now. In particular, x there meant an arrow from strand 1 to strand 3, while here it means a one-wheel on (capped) strand 1, and similarly for y. Fortunately, there is a linear map σ: A^{sw}(↑_{3}) → A^{sw}(_{2}), where σ “closes the third strand and turns it into a chord (or internal) strand, and caps the first two strands”, as shown on the right. This map is well defined (in fact, it kills almost all relations, and turns one \overrightarrow{ST U} into an \overrightarrow{IHX}). Under this map, using our abusive notation, σ(x) = x and σ(y) = y.

Now we can apply Equation (12) to get e^{uD}e^{c(x+y)}e^{−uD} = e^{c(log e^{y}e^{x})}. Substituting this into the cap equation we obtain e^{b}e^{c(log e^{y}e^{x})} = e^{c(x)+c(y)}, which, using that tails commute, implies b = c(x) + c(y) − c(log e^{y}e^{x}). Switching strands 1 and 2, we obtain

b^{21} = c(x) + c(y) − c(log e^{x}e^{y})  (14)

In summary, we can use (V, C) to produce F := e^{D^{21}} (sorry^{14}) and a := −2c which satisfy the Alekseev-Torossian equations (11), as follows: e^{D^{21}} acts on 𝔩𝔦𝔢_{2} by conjugation by e^{uD^{21}}, so the first part of (11) is implied by (12). The second half of (11) is true due to (13) and (14).

On the other hand, suppose that we have found F ∈ TAut_{2} and even Duflo function a ∈ tr_{1} satisfying (11). Then set D^{21} := log F, b^{21} := \frac{−j(e^{D^{21}})}{2}, and c ∈ δ̃^{−1}(b^{21}), in particular c = −\frac{a}{2} works. Then V = e^{b}e^{uD} and the even cap value C = e^{c} satisfy the equations for homomorphic expansions (R4), (U) and (C), and hence define a homomorphic expansion of wTF with W = 1.

Furthermore, these maps between solutions of the KV problem and nv-small homomorphic expansions for wTF with W = 1 are obviously inverses of each other, and hence they provide a bijection between these sets as stated. □

Remark 4.18. The fact that Z can be chosen to have W = 1 and C even follows from Proposition 6.2 of [AT]. In Proposition 6.2 Alekseev and Torossian show that the even part of f is \frac{1}{2} \frac{log(e^{x/2}−e^{−x/2})}{x}, and that for any f with this even part (and any odd part) there exists a corresponding solution F of the generalized KV problem. In particular, f can be assumed to be even, and hence it can be guaranteed that C consists of even wheels only.

4.6. Orientable w-tangled foams. There is a sub-circuit algebra of wTF consisting of the w-tangled foams which contain no wens. We call this the circuit algebra of orientable w-foams, and denote it by wTF^{o}. (These foams can be equipped with a global surface orientation, which induces crossing and vertex signs consistent with the signs suggested by the diagrams. However, this is not necessary.)

Lemma 4.19. Let F ∈ wTF be a w-foam with the property that there are an even number of wens along any path connecting two tangle ends, and along any cycle in F. Then all of the wens in F cancel by the wen relations. Furthermore, the process of cancelling all wens can be made canonical by a choice – for each connected component of the skeleton of F – of a spanning tree T, and a basepoint on T, which is a tangle end if there are any.

14We apologize for the annoying 2 ↔ 1 transposition in this equation, which makes some later equations, especially (19), uglier than they could have been. There is no depth here, just mis-matching conventions between us and Alekseev-Torossian.

Proof. First note that the statement of the lemma concerns only the skeleton σ(F): by the FR relations wens slide through crossings, at the possible cost of more virtual crossings. The skeleton of F is a uni-trivalent graph whose univalent ends are either caps of tangle ends. Due to the CW relation, capped edges can be ignored, that is, deleted without loss of generality. Thus, assume that σ(F) is a uni-trivalent graph in which all univalent vertices are tangle ends.

Given a choice of spanning tree T for σ(F) and a base point on it, there is a unique way to “clear T of wens”. Namely, use the TV relation to push wens off of T away from the base point. The T V relation does not change the parity of the number of wens along any cycle, or any path connecting two tangle ends. At the end of this process, all wens will end up either on an edge of σ(F) not in T, or at a tangle end (which are all necessarily in T).

At the end of the process, there is still an even number of wens on the path from any given tangle end to the base point (which is also necessarily a tangle end in this case), and so there is an even number of wens at each tangle end, therefore they cancel by the W^{2} relation. For any non-T edge e of σ(F), there is a unique path γ in T which connects the two ends of e. Since there originally was an even number of wens along the cycle e ∪ γ, there is an even number of wens on e at the end of the process, which therefore cancel. □

We derive a generators - relations - operations presenation for wTF^{o}. Since the wen is no longer a generator, there are no wen relations. The operations S_{e}, u_{e} and d_{e} restrict to wTF^{o}. The composition with wens in wTF induces an involution on wTF^{o}: while wens are not included in wTF^{o}, composition with a wen at every tangle end is well-defined:

Definition 4.20. For F ∈ wTF^{o}, consider F as an element of wTF, and let F̄ denote F composed with a wen at every tangle end. Then by Lemma 4.19, F̄ ∈ wTF^{o}. We call this operation wenjugation, and denote it by − : wTF^{o} → wTF^{o}.

Definition 4.21. The circuit algebra of oriented w-foams is defined by the presentation

wTF^{o} = CA ⟨ , , , , | R1^{s}, R2, R3, R4, OC, CP | S_{e}, u_{e}, d_{e}, − ⟩ .

Next, we verify that wTF^{o} – as defined by the presentation above – injects into wTF. In other words, the generators and relations description above is indeed a description of sub-circuit algebra of wTF generated by all orientable (non-wen) wTF generators.

Proposition 4.22. The circuit algebra of oriented w-foams wTF^{o} injects into wTF.

Proof. We need to show that given F, F′ ∈ wTF^{o} for which F F′ via a sequence of wTF relations, then F F′ also in wTF^{o}. This can be verified explicitly, as follows. Choose a spanning tree and base point for each connected component of σ(F) = σ(F′). Let F = F_{0} ∼ F_{1} ∼ ⋯ ∼ F_{n} = F′ be a sequence of wTF moves. Since all F_{i} are wTF -equivalent to F ∈ wTF^{o}, they all satisfy the conditions of Lemma 4.19. Via the process of Lemma 4.19, each F_{i} (i=0,…, n) is canonically equivalent to an element of wTF^{o}, call this element Ω(F_{i}). Hence, we only need to show that Ω(F_{i}) ∼ Ω(F_{i+1}) in wTF^{o}, where F_{i} and F_{i+1} differ in a single relation in wTF. This is obvious if that relation is not a wen relation, easy for the W^{2} and CW relations, and directly verified with some effort for the F R and T V relations. □

The circuit algebra wTF^{o} is again skeleton graded, with skeleton circuit algebra given by

S^{0} = CA⟨ , , ⟩

The associated graded structure – which we continue to denote by A^{sw} to avoid too many superscripts – consists of arrow diagrams on uni-coloured skeleta (elements of S^{o}), given by the presentation

A^{sw} = CA ⟨ , , , | \overrightarrow{4T}, TC, VI, CP, RI | S_{e}, u_{e}, d_{e}, − ⟩ .

We denote by − : A^{sw} → A^{sw} the associated graded operation of wenjugation. It is also an involution, and coincides with the operation D ↦ \overline{D} defined in Section 4.4. Namely, \overline{D} is the arrow diagram D multiplied with (−1)^{#{arrow tails}}.

As before, arrow diagrams have an alternative, equivalent description in terms of Jacobi diagrams, as in Theorem 4.4.

The main theorem of this section states that homomorphic expansions for wTF^{o} are in bijection with Kashiwara-Vergne solutions, without restriction on the Duflo function:

Theorem 4.23. There exist a group-like homomorphic expansions for wTF^{o}, and there is a bijection between the set of solutions (F, a) of the generalized KV equations (11) and the set of v-small group-like homomorphic expansions for wTF^{o}.

Proof. Since there are no wens, a homomorphic expansion is determined by the values R, C, and V, with Z(�) = (R^{−1})^{21} by the R2 relation, and Z( ) = V^{−1} by the homomorphicity with respect to edge unzip.

We derive R = e^{a}, and the (R4) and (C) equations as before: from the R3, R4 and CP relations, and the homomorphicity with respect to S_{e}, u_{e} and d_{e}. There are no wen relations, hence no restriction on the odd part of C, nor a Unitarity equation.

Recall that for wTF, the TV relation gave rise to the unitarity equation. Since one side of the TV relation is the wenjugate of the vertex . Thus, homomorphicity with respect to wenjugation is equivalent to the Unitarity equation (U).

We showed in the proof or Theorem 4.8 that the equations (R4), (C) and (U), given the v-small condition, translate exactly to the Kashiwara–Vergne equations. This completes the proof. □

4.7. Interlude: u-Knotted Trivalent Graphs. The “usual”, or classical knot-theoretical objects corresponding to wTF are loosely speaking Knotted Trivalent Graphs, or KTGs. We give a brief introduction/review of this structure before studying the relationship between their homomorphic expansions and homomorphic expansions for wTF. The last goal of this paper is to show that the topological relationship between the two spaces explains the relationship between the KV problem and Drinfel’d associators.

A trivalent graph is a graph with three edges meeting at each vertex, equipped with a cyclic orientation of the three half-edges at each vertex. KTGs are framed embeddings of trivalent graphs into ℝ^{3}, regarded up to isotopies. The skeleton of a KTG is the trivalent graph (as a combinatorial object) behind it. For a detailed introduction to KTGs see for example [BND]. Here we only recall the most important facts. The reader might recall that in Section 3, the w-knot section, of [WKO1] we only dealt with long w-knots, as the w-theory of round knots is essentially trivial (see [WKO1, Theorem 3.17]). A similar issue arises with “w-knotted trivalent graphs”. Hence, the space we are really interested in is “long KTGs”, meaning, trivalent tangles with 1 or 2 ends.

S^{T}^{,} ^{S}^{T}  ^{,}^{T}  ^{S}  ^{T}  ^{S} γ  ^{e}  u_{e}(γ)^{stick-on e}^{insert in e}^{unzip e}^{e}^{e}

Long KTGs form an algebraic structure with operations as follows. Orientation switch reverses the orientation of a specified edge. Edge unzip doubles a specified edge as shown on the right. Tangle insertion is inserting a small copy of a (1, 1)-tangle S into the middle of some specified edge of a tangle T, as shown in the second row on the right (tangle composition is a special case of this). The stick-on operation “sticks a 1-tangle S onto a specified edge of another tangle T ”, as shown. (In the figures T is a 2-tangle, but this is irrelevant.) Disjoint union of two 1-tangles produces a 2-tangle. Insertion, disjoint union and stick-on are a slightly weaker set of operations than the connected sum of [BND].

The associated graded structure of the algebraic structure of long KTGs is the graded space A^{u} of chord diagrams on trivalent graph skeleta, modulo the 4T and vertex invariance (VI) relations. The induced operations on A^{u} are as expected: orientation switch multiplies a chord diagram by (−1) to the number of chord endings on the edge. The edge unzip u_{e} maps a chord diagram with k chord endings on the edge e to a sum of 2^{k} diagrams where each chord ending has a choice between the two daughter edges. Finally, tangle insertion, stick-on and disjoint union induces the insertion, sticking on and disjoint union of chord diagrams, respectively.

u(ν^{1/2}) ν^{−1/2} ν^{−1/2} Z^{old}(γ) Z^{old}(u(γ))

In [BND] the authors prove that there is no homomorphic expansion for KTGs. This theorem, as well as the proof, applies to long KTGs with slight modifications. However there are wellknown — and nearly homomorphic — expansions constructed by extending the Kontsevich integral to KTGs, or from Drinfel’d associators. There are several such constructions ([MO], [CL], [Da]). For now, let us denote any one of these expansions by Z^{old}. All Z^{old} are “almost homomorphic”: they intertwine every operation except for edge unzip with their chord-diagrammatic counterparts; but commutativity with unzip fails by a controlled amount, as shown on the right. Here ν denotes the “invariant of the unknot”, the value of which was conjectured in [BGRT] and proven in [BLT].

In [BND] the authors fix this anomaly by slightly changing the space of KTGs and adding some extra combinatorics (“dots” on the edges), and construct a homomorphic expansion for this new space by a slight adjustment of Z^{old}. Here we are going to use a similar but different adjustment of the space of trivalent 1- and 2-tangles. Namely we break the symmetry of the vertices and restrict the set of allowed unzips.

Definition 4.24. A “signed KTG”, denoted sKTG, is a trivalent oriented 1- or 2-tangle embedded in ℝ^{3} with a cyclic orientation of edges meeting at each vertex, and in addition each vertex is equipped with a sign and one of the three incident edges is marked as distinguished (sometimes denoted by a thicker line). Our pictorial convention will be that a vertex drawn in a “ Y ” shape with all edges oriented up and the top edge distinguished is always positive and a vertex drawn in a “Y ” shape with edges oriented up and the bottom edge distinguished is always negative (see Figure 20).

T  S  T  ^{S} ,, T  S  T  ^{S}^{stick-on}  ^{+}^{stick-on}  ^{−}

The algebraic structure sKTG has one kind of objects for each skeleton (a skeleton is a uni-trivalent graph with signed vertices but no embedding), as well as several operations: orientation switch, edge unzip, tangle insertion, disjoint union of 1-tangles, and stick-on. Orientation switch of either of the non-distinguished edges changes the sign of the vertex, switching the orientation of the distinguished edge does not. Unzip of an edge is only allowed if the edge is distinguished at both of its ends and the vertices at either end are of opposite signs. The stick-on operation can be done in either one of the two ways shown on the right (i.e., the stuck-on edge can be attached at a vertex of either sign, but it can not become the distinguished edge of that vertex).

To consider expansions of sKTG, and ultimately the compatibility of these with Z^{w}, we first note that sKTG is finitely generated (and therefore any expansion Z^{u} is determined by its values on finitely many generators). The proof of this is not hard but somewhat lengthy, so we postpone it to Section 5.2.

Proposition 4.25. The algebraic structure sKTG is finitely generated by the following list of elements:

,^{+}^{−}  ,^{−}^{+}  ,^{+}^{−}^{right}^{twist} ^{left}^{twist}^{strand}  ^{+}^{+}^{−}^{−} ,^{right}^{associator}  ^{left}^{associator}^{+}^{−}^{−}^{+} ,^{bubble}  ^{balloon} ^{noose}^{+} ,  ^{−} ,

Note that we ignore edge orientations for simplicity in the statement of this proposition; this is not a problem as orientation switches are allowed in sKTG without restriction.

4.7.1. Homomorphic expansions for sKTG. Suppose that Z^{u} : sKTG → A^{u} is a homomorphic expansion. We hope to determine the value of Z^{u} on each of the generators.

u

The value of Z^{u} on the single strand is an element of A^{u}(↑) whose square is itself, hence it is 1. The value of the bubble is an element x ∈ A^{u}(↑_{2}), as all chords can be pushed to the “bubble” part using the VI relation. Two bubbles can be composed and unzipped to produce a single bubble (see on the right), hence we have x^{2} = x, which implies x = 1 in A^{u}(↑_{2}).

Recall that a Drinfel’d associator is a group-like element Φ ∈ A^{u}(↑_{3}) along with a grouplike element R^{u} ∈ A^{u}(↑_{2}) satisfying the so-called pentagon and positive and negative hexagon equations, as well as a non-degeneracy and mirror skew-symmetry property. For a detailed explanation see Section 4 of [BND]; associators were first defined in [Dr2]. We claim that the Z^{u}-value Φ of the right associator, along with the value R^{u} of the right twist forms a Drinfel’d associator pair. The proof of this statement is the same as the proof of Theorem 4.2 of [BND], with minor modifications (making heavy use of the assumption that Z^{u} is homomorphic). It is easy to check by composition and unzips that the value of the left associator and the left twist are Φ^{−1} and (R^{u})^{−1}, respectively. Note that if Φ is a horizontal chord associator (i.e., all the chords of Φ are horizontal on three strands) then R^{u} is forced to be e^{c/2} where c denotes a single chord. Note that the reverse is not true: there exist non-horizontal chord associators Φ that satisfy the hexagon equations with R^{u} = e^{c/2}.

n · b =  Φ  −1

Let b and n denote the Z^{u}-values of the balloon and the noose, respectively. Note that using the V I relation all chord endings can be pushed to the “looped” strands, so b and n live in A^{u}(↑), as seen in Figure 18. The argument in that figure shows that n · b is the inverse in A^{u}(↑) of “an associator on a squiggly strand”, as shown on the right. In Figure 18 we start with the sKTG on the top left and either apply Z^{u} followed by unzipping the edges marked by stars, or first unzip the same edges and then apply Z^{u}. Since Z^{u} is homomorphic, the two results in the bottom right corner must agree. (Note that two of the four unzips we perform are “illegal”, as the strand directions can’t match. However, it is easy to get around this issue by inserting small bubbles at the top of the balloon and the bottom of the noose, and switching the appropriate edge orientations before and after the unzips. The Z^{u}-value of a bubble is 1, hence this will not effect the computation and so we ignore the issue for simplicity.)

Z^{u} n  Φ Z^{u} b  n Φ b

Figure 18. Unzipping a noose and a balloon to a squiggle.

close up stick on  Z^{old}  ·ν^{−1} remove extras Konts. integral ν  ν  ν

Figure 19. Computing the Z^{old} value of the noose. The third step uses that the Kontsevich integral of KTGs is homomorphic with respect to the “connected sum” operation and that the value of the unknot is ν (see [BND] for an explanation of both of these facts).

In addition, it follows from Theorem 4.2 of [BND] via deleting two edges that the inverse of an “associator on a squiggly strand” is ν, the invariant of the unknot. To summarize, we have proven the following:

Lemma 4.26. If Z^{u} is a homomorphic expansion then the Z^{u} values of the strand and the bubble are 1, the values of the right associator and right twist form an associator pair (Φ, R^{u}), and the values of the left twist and left associator are inverses of these. With n and b denoting the value of the noose and the balloon, respectively, and ν being the invariant of the unknot, we have n · b = ν in A^{u}(↑).

The natural question to ask is whether any triple (Φ, R^{u}, n) gives rise to a homomorphic expansion. We don’t know whether this is true, but we do know that any pair (Φ, R^{u}) gives rise to a “nearly homomorphic” expansion of KTGs [MO, CL, Da], and we can construct a homomorphic expansion for sKTG from any of these (as shown below). However, all of these expansions take the same specific value on the noose and the balloon (also see below). We don’t know whether there really is a one parameter family of homomorphic expansions Z^{u} for each choice of (Φ, R^{u}) or if we are simply missing a relation.

Z^{old} ^{T} := ν^{−1} · Z^{old}  ^{T}

We now construct explicit homomorphic expansions Z^{u}: sKTG → A^{u} from any Z^{old} (where Z^{old} stands for an “almost homomorphic” expansion of KTGs) as follows. First of all we need to interpret Z^{old} as an invariant of 2-tangles. This can be done by connecting the top and bottom ends by a non-interacting long strand followed by a normalization, as shown on the right. By “multiplying by ν^{−1}” we mean that after computing Z^{old} we insert ν^{−1} on the long strand (recall that ν is the “invariant of the unknot”). We interpret Z^{old} of a 1-tangle as follows: stick the 1-tangle onto a single strand to obtain a 2-tangle, then proceed as above. The result will only have chords on the 1-tangle (using that the extensions of the Kontsevich Integral are homomorphic with respect to “connected sums”), so we define the result to be the value of Z^{old} on the 1-tangle. As an example, we compute the value of Z^{old} for the noose in Figure 19 (note that the computation for the balloon is the same).

+ e^{c/4} ν^{−1/4}ν^{1/4} ν^{−1/4}  −  − e^{−c/4} ν^{1/4} ν^{−1/4} ν^{−1/4} +

Figure 20. Normalizations for Z^{u} at the vertices.

b = n = ^{e^{−c/4}}^{ν^{1/2}}^{e^{c/4}}^{ν^{1/2}}

Now to construct a homomorphic Z^{u} from Z^{old} we add normalizations near the vertices, as in Figure 20, where c denotes a single chord. Checking that Z^{u} is a homomorphic expansion is a simple calculation using the almost homomorphicity of Z^{old}, which we leave to the reader. The reader can also verify that Z^{u} of the strand and the bubble is 1 as it should be. Z^{u} of the right twist is e^{c/2} and Z^{u} of the right associator is a Drinfel’d associator Φ (note that Φ depends on which Z^{old} was used). From the calculation of Figure 19 it follows that the Z^{u} value of the balloon and the noose (for any Z^{old}) are as shown on the right, and indeed n · b = ν.

4.8. The relationship between sKTG and wTF^{o}. We move on to the question of compatibility between the homomorphic expansions Z^{u} and Z^{w} (from now on we are going to refer to the homomorphic expansion of wTF^{o} — called Z in the previous section — as Z^{w} to avoid confusion).

There is a map a: sKTG → wTF^{o}, given by interpreting sKTG diagrams as wTF^{o} diagrams. In particular, positive vertices (of edge orientations as shown in Figure 20) are interpreted as the wTF^{o} vertex and negative vertices as the wTF^{o} vertex . (The map a can also be interpreted topologically as Satoh’s tubing map.) The induced map α: A^{u} → A^{sw} is as defined in Section 3.3, that is, α maps each chord to the sum of its two possible orientations. Hence we can ask whether the two expansions are compatible (or can be chosen to be compatible), which takes us to the main result of this section:

\xymatrix{sKTG \ar[d]^{Z^{u}} \ar[r]^{a} & wTF^{o} \ar[d]^{Z^{w}} \\ A^{u} \ar[r]^{α} & A^{sw}} (15)

Theorem 4.27. Let Z^{u} be a homomorphic expansion for sKTG with the properties that Φ is a horizontal chord associator and n = e^{−c/4}ν^{1/2} in the sense of Section 4.7.1.^{15} Then there exists a homomorphic expansion Z^{w} for wTF^{o} compatible with Z^{u} in the sense that the square on the right commutes.

Furthermore, such Z^{w} are in one to one correspondence^{16} with “symmetric solutions of the KV problem” satisfying the KV equations (11), the “twist equation” (17) and the associator equation (19).

15It will become apparent that in the proof we only use slightly weaker but less aesthetic conditions on Z^{u}.

16An even nicer theorem would be a classification of homomorphic expansions for the combined algebraic structure (sKTG ⟶^{a} wTF^{o}) in terms of solutions of the KV problem. The two obstacles to this are clarifying whether there is a free choice of n for Z^{u}, and — probably much harder — how much of the horizontal chord condition is necessary for a compatible Z^{w} to exist.

Before moving on to the proof let us state and prove the following Lemma, to be used repeatedly in the proof of the theorem.

Lemma 4.28. If a and b are group-like elements in A^{sw}(↑_{n}), then a = b if and only if π(a) = π(b) and aa^{∗} = bb^{∗}. Here π is the projection induced by π: P^{w}(↑_{n}) → 𝔱𝔡𝔢𝔯_{n} ⊕𝔞_{n} (see Section 3.2), and ∗ refers to the adjoint map of Definition 3.26.

Proof. Write a = e^{w}e^{uD} and b = e^{w′}e^{uD′}, where w ∈ 𝔱𝔯_{n}, D ∈ 𝔱𝔡𝔢𝔯_{n} ⊕𝔞_{n} and u: 𝔱𝔡𝔢𝔯_{n} ⊕𝔞_{n} → P_{n} is the “upper” map of Section 3.2. Assume that π(a) = π(b) and aa^{∗} = bb^{∗}. Since π(a) = e^{D} and π(b) = e^{D′}, we conclude that D = D′. Now we compute aa^{∗} = e^{w}e^{uD}e^{−lD}e^{w} = e^{w}e^{j(D)}e^{w}, where j: 𝔱𝔡𝔢𝔯_{n} → 𝔱𝔯_{n} is the map defined in Section 5.1 of [AT] and discussed in 3.27 of this paper. Now note that both w and j(D) are elements of 𝔱𝔯_{n}, hence they commute, so aa^{∗} = e^{2w+j(D)}. Thus, aa^{∗} = bb^{∗} means that e^{2w+j(D)} = e^{2w′+j(D)}, which implies that w = w′ and a = b. □

Proof of Theorem 4.27. In addition to being a homomorphic expansion for wTF^{o}, Z^{w} has to satisfy an the added condition of being compatible with Z^{u}. Since sKTG is finitely generated, this translates to one additional equation for each generator of sKTG, some of which are automatically satisfied. To deal with the others, we use the machinery established in the previous sections to translate these equations to conditions on F, and they turn out to be the properties studied in [AT] which link solutions of the KV problem with Drinfel’d associators.

To start, note that for the single strand and the bubble the commutativity of the square (15) is satisfied with any Z^{w}: both the Z^{u} and Z^{w} values are 1 (note that the Z^{w} value of the bubble is 1 due to the unitarity (U) of Z^{w}). Each of the other generators will require more study.

Commutativity of (15) for the twists. Recall that the Z^{u}-value of the right twist (for a Z^{u} with horizontal chord Φ) is R^{u} = e^{c/2}; and note that its Z^{w}-value is V^{−1}RV^{21}, where R = e^{a_{12}} is the Z^{w}-value of the crossing (and a_{12} is a single arrow pointing from strand 1 to strand 2). Hence the commutativity of (15) for the right twist is equivalent to the “Twist Equation” α(R^{u}) = V^{−1}RV^{21}. By definition of α, α(R^{u}) = e^{\frac{1}{2} (a_{12}+a_{21})}, where a_{12} and a_{21} are single arrows pointing from strand 1 to 2 and 2 to 1, respectively. Hence we have

e^{\frac{1}{2} (a_{12}+a_{21})} = V^{−1}RV^{21}.  (16)

To translate this to the language of [AT], we use Lemma 4.28, which implies that it is enough for V to satisfy the Twist Equation “on tree level” (i.e., after applying π), and for which the adjoint condition of the Lemma holds.

We first prove that the adjoint condition holds for any homomorphic expansion of wTF^{o}. Multiplying the left hand side of the Twist Equation by its adjoint, we get

e^{\frac{1}{2} (a_{12}+a_{21})}(e^{\frac{1}{2} (a_{12}+a_{21})})^{∗} = e^{\frac{1}{2} (a_{12}+a_{21})}e^{− \frac{1}{2} (a_{12}+a_{21})} = 1.

As for the right hand side, we have to compute V^{−1}RV^{21}(V^{21})^{∗}R^{∗}(V^{−1})^{∗}. Since V is unitary (Equation (U)), V V^{∗} = 1. Now R = e^{a_{12}}, so R^{∗} = e^{−a_{12}} = R^{−1}, hence the expression on the right hand side also simplifies to 1, as needed.

As for the “tree level” of the Twist Equation, recall that in Section 4.4 we used Alekseev and Torossian’s solution F ∈ TAut_{2} to the Kashiwara–Vergne equations [AT] to find solutions V to equations (R4),(U) and (C). We produced V from F by setting F = e^{D^{21}} with D ∈ 𝔱𝔡𝔢𝔯_{2}^{s},

1  ^{2}  ^{3} Using V I to push to the middle three strands.^{V} ^{V}^{V^{−}}^{V^{−}}

Figure 21. The Z^{w}-value of the right associator.

b := \frac{−j(F)}{2} ∈ 𝔱𝔯_{2} and V := e^{b}e^{uD}, so F is “the tree part” of V, up to re-numbering strands. Hence, the tree level Twist Equation translates to a new equation for F. Substituting V = e^{b}e^{uD} into the Twist Equation we obtain e^{\frac{1}{2} (a_{12}+a_{21})} = e^{−uD}e^{−b}e^{a_{12}}e^{b^{21}}e^{uD^{21}}, and applying π, we get

e^{\frac{1}{2} (a_{12}+a_{21})} = (F^{21})^{−1}e^{a_{12}}F.  (17)

In [AT] the solutions F of the KV equations which also satisfy this equation are called “symmetric solutions of the Kashiwara-Vergne problem” discussed in Sections 8.2 and 8.3. (Note that in [AT] R denotes e^{a_{21}}).

Commutativity of (15) for the associators. Recall that the Z^{u} value of the right associator is a Drinfel’d associator Φ ∈ A^{u}(↑_{3}); for the Z^{w} value see Figure 21. Hence the new condition on V is the following:

α(Φ) = V_{−}^{(12)3}V_{−}^{12}V^{23}V^{1(23)} in A^{sw}(↑_{3})  (18)

Again we treat the “tree and wheel parts” separately using Lemma 4.28. As Φ is by definition group-like, let us denote Φ =: e^{ϕ}. We first verify that the “wheel part” or adjoint condition of the Lemma holds. Starting with the right hand side of Equation (18), the unitarity V V^{∗} = 1 of V implies that

V_{−}^{(12)3}V_{−}^{12}V^{23}V^{1(23)}(V^{1(23)})^{∗}(V^{23})^{∗}(V_{−}^{12})^{∗}(V_{−}^{(12)3})^{∗} = 1.

For the left hand side of (18) we need to show that e^{α(ϕ)}(e^{α(ϕ)})^{∗} = 1 as well, and this is true for any horizontal chord associator. Indeed, restricted to the α-images of horizontal chords ∗ is multiplication by −1, and as it is an anti-Lie morphism, this fact extends to the Lie algebra generated by α-images of horizontal chords. Hence e^{α(ϕ)}(e^{α(ϕ)})^{∗} = e^{α(ϕ)}e^{α(ϕ)^{∗}} = e^{α(ϕ)}e^{−α(ϕ)} = 1.

On to the tree part, applying π to Equation (18) and keeping in mind that V_{−} = V^{−1} by the unitarity of V, we obtain

e^{πα(ϕ)} = (F^{3(12)})^{−1}(F^{21})^{−1}F^{32}F^{(23)1} = e^{−D^{(12)3}}e^{−D^{12}}e^{D^{23}}e^{D^{1(23)}} in SAut_{3} := exp(𝔰𝔡𝔢𝔯_{3}) ⊂ TAut_{3} . (19)

This is Equation (26) of [AT], up to re-numbering strands 1 and 2 as 2 and 1^{17}. The following fact from [AT] (their Theorem 7.5, Propositions 9.2 and 9.3 combined) implies that there is a solution F to the KV equations (11) which also satisfies (17) and (19).

Fact 4.29. If Φ′ = e^{ϕ′} is an associator in SAut_{3} so that j(Φ′) = 0^{18} then Equation (19) has a solution F = e^{D^{21}} which is also a solution to the KV equations, and all such solutions are symmetric (i.e. verify the Twist Equation (17)). □

To use this Fact, we need to show that Φ′ := πα(Φ) is an associator in SAut_{3} and that j(Φ′) = j(πα(Φ)) = 0. The latter is the unitarity of Φ which is already proven. The former follows from the fact that Φ is an associator and the fact (Theorem 3.28) that the image of πα is contained in 𝔰𝔡𝔢𝔯 (ignoring degree 1 terms, which are not present in an associator anyway).

In summary, the condition of the Fact are satisfied and so there exists a solution F which in turn induces a Z^{w} which is compatible with Z^{u} for the strand, the bubble, the twists and the associators. That is, all generators of sKTG except possibly the balloon and the noose. As the last step of the proof of Theorem 4.27 we show that any such Z^{w} also automatically make (15) commutative for the balloon and the noose.

Z^{w}  ^{=} ^{S_{1}(V)}

Commutativity of (15) for the balloon and the noose. Since we know the Z^{u}-values B and n of the balloon and the noose, we start by computing Z^{w} of the noose. Z^{w} assigns a V value to the vertex with the first strand orientation switched as shown in the figure on the right. The balloon is the same, except with the inverse vertex and the second strand reversed. Hence what we need to show is that the two equations below hold:

= S_{1}(V)  ^{α(ν)^{1/2}} e^{\frac{−D_{A}}{2}}  e^{\frac{D_{A}}{2}}^{α(ν)^{1/2}} S_{2}(V^{−1})

Let us denote the left hand side of the first and second equation above by n^{w} and b^{w}, respectively (that is, the Z^{w} value of the noose and the balloon, respectively). We start by proving that the product of these two equations holds, namely that n^{w}b^{w} = α(ν). (We used that any local (small) arrow diagram on a single strand is central in A^{sw}(↑_{n}), hence the cancellations.) This product equation is satisfied due to an argument identical to that of Figure 18, but carried out in wTF^{o}, and using that by the compatibility with associators, Z^{w} of an associator is α(Φ).

What remains is to show that the noose and balloon equations hold individually. In light of the results so far, it is sufficient to show that

n^{w} = b^{w} · e^{−D_{A}},  (20)

17Note that in [AT] “Φ′ is an associator” means that Φ′ satisfies the pentagon equation, mirror skewsymmetry, and positive and negative hexagon equations in the space SAut_{3}. These equations are stated in [AT] as equations (25), (29), (30), and (31), and the hexagon equations are stated with strands 1 and 2 re-named to 2 and 1 as compared to [Dr2] and [BND]. This is consistent with F = e^{D^{21}}.

18The condition j(ϕ′) = 0 is equivalent to the condition Φ ∈ KRV_{3}^{0} in [AT]. The relevant definitions in [AT] can be found in Remark 4.2 and at the bottom of page 434 (before Section 5.2).

u Z^{w}  u Z^{w}  ^{S_{1}(V)}^{C}^{S_{2}(V^{−1})}^{C}  ^{S_{2}(V^{−1})}^{S_{1}(V)}^{C}^{S(C)} ^{C}^{S(C)}

Figure 22. The proof of Equation (21). Note that the unzips are “illegal”, as the strand directions don’t match. This can be fixed by inserting a small bubble at the bottom of the noose and doing a number of orientation switches. As this doesn’t change the result or the main argument, we suppress the issue for simplicity. Equation (21) is obtained from this result by multiplying by S(C)^{−1} on the bottom and by C^{−1} on the top.

where D_{A} stands for a single arrow on one strand (whose direction doesn’t matter due to the RI relation. As stated in [WKO1, Theorem 3.15], A^{sw}(↑_{1}) is the polynomial algebra freely generated by the arrow D_{A} and wheels of degrees 2 and higher. Since V is group-like, n^{w} (resp. b^{w}) is an exponential e^{A_{1}} (resp. e^{A_{2}}) with A_{1}, A_{2} ∈ A^{sw}(↑_{1}). We want to show that e^{A_{1}} = e^{A_{2}} · e^{−D_{A}}, equivalently that A_{1} = A_{2} − D_{A}.

In degree 1, this can be done by explicit verification. Let A_{1}^{≥2} and A_{2}^{≥2} denote the degree 2 and higher parts of A_{1} and A_{2}, respectively. We claim that capping the strand at both its top and its bottom takes e^{A_{1}} to e^{A_{1}^{≥2}}, and similarly e^{A_{2}} to e^{A_{2}^{≥2}}. (In other words, capping kills arrows but leaves wheels un-changed.) This can be proven similarly to the proof of Lemma 4.5, but using

F′ := ∑_{k_{1},k_{2}=0}^{∞} \frac{(−1)^{k_{1}+k_{2}}}{k_{1}!k_{2}!} D_{A}^{k_{1}+k_{2}}S_{L}^{k_{1}}S_{R}^{k_{2}}

in place of F in the proof. What we need to prove, then, is the following equality, and the proof is shown in Figure 22.

S_{1}(V) _{=} _{S_{2}(V^{−1})} _{.}  (21)

This concludes the proof of Theorem 4.27. □

Recall from Section 3.3 that there is no commutative square linking Z^{u}: uT → A^{u} and Z^{w}: wT → A^{sw}, for the simple reason that the Kontsevich integral for tangles Z^{u} is not canonical, but depends on a choice of parenthesizations for the “bottom” and the “top” strands of a tangle T. Yet given such choices, a tangle T can be “closed up with trees” as within the proof of Proposition 4.25 (see Section 5) into an sKTG which we will denote G. For G a commutativity statement does hold as we have just proven. The Z^{u} and Z^{w} invariants of T and of G differ only by a number of vertex-normalizations and vertex-values on skeleton-trees at the bottom or at the top of G, and using VI, these values can slide so they are placed on the original skeleton of T. This is summarized as the following proposition:

Proposition 4.30. Let n and n′ be natural numbers. Given choices c and and c′ of parenthesizations of n and n′ strands respectively, there exists invertible elements C ∈ A^{sw}(↑_{n}) and C′ ∈ A^{sw}(↑_{n′}) so that for any u-tangle T with n “bottom” ends and n′ “top” ends we have

αZ_{c,c′}^{u}(T) = C^{−1}Z^{w}(aT)C′,

where Z_{c,c′}^{u} denotes the usual Kontsevich integral of T with bottom and top parenthesizations c and c′.

For u-braids the above proposition may be stated with c = c′ and then C and C′ are the same.

5. Odds and Ends

5.1. Motivation for circuit algebras: electronic circuits. Electronic circuits are made of “components” that can be wired together in many ways. On a logical level, we only care to know which pin of which component is connected with which other pin of the same or other component. On a logical level, we don’t really need to know how the wires between those pins are embedded in space (see Figures 23 and 24). “Printed Circuit Boards” (PCBs) are operators that make smaller components (“chips”) into bigger ones (“circuits”) — logically speaking, a PCB is simply a set of “wiring instructions”, telling us which pins on which components are made to connect (and again, we never care precisely how the wires are routed provided they reach their intended destinations, and ever since the invention of multi-layered PCBs, all conceivable topologies for wiring are actually realizable). PCBs can be composed (think “plugging a graphics card onto a motherboard”); the result of a composition of PCBs, logically speaking, is simply a larger PCB which takes a larger number of components as inputs and outputs a larger circuit. Finally, it doesn’t matter if several PCB are connected together and then the chips are placed on them, or if the chips are placed first and the PCBs are connected later; the resulting overall circuit remains the same.

Figure 23. The J-K flip flop, a very basic memory cell, is an electronic circuit that can be realized using 9 components — two triple-input “and” gates, two standard “nor” gates, and 5 “junctions” in which 3 wires connect (many engineers would not consider the junctions to be real components, but we do). Note that the “crossing” in the middle of the figure is merely a projection artifact and does not indicate an electrical connection, and that electronically speaking, we need not specify how this crossing may be implemented in ℝ^{3}. The J-K flip flop has 5 external connections (labelled J, K, CP, Q, and Q’) and hence in the circuit algebra of computer parts, it lives in C_{5}. In the directed circuit algebra of computer parts it would be in C_{3,2} as it has 3 incoming wires (J, CP, and K) and two outgoing wires (Q and Q’).

5.2. Proof of Proposition 4.25. We are going to ignore strand orientations throughout this proof for simplicity. This is not an issue as orientation switches are allowed in sKTG without restriction. We are also going to omit vertex signs from the pictures given the pictorial convention stated in Section 4.7.

Figure 24. The circuit algebra product of 4 big black components and 1 small black component carried out using a green wiring diagram, is an even bigger component that has many golden connections (at bottom). When plugged into a yet bigger circuit, the CPU board of a laptop, our circuit functions as 4,294,967,296 binary memory cells.

We need to prove that any sKTG (call it G) can be built from the generators listed in the statement of the proposition, using sKTG operations. To show this, consider a Morse drawing of G, that is, a planar projection of G with a height function so that all singularities along the strands are Morse and so that every “feature” of the projection (local minima and maxima, crossings and vertices) occurs at a different height.

The idea in short is to decompose G into levels of this Morse drawing where at each level only one “feature” occurs. The levels themselves are not sKTG’s, but we show that the composition of the levels can be achieved by composing their “closed-up” sKTG versions followed by some unzips. Each feature gives rise to a generator by “closing up” extra ends at its top and bottom. We then show that we can construct each level using the generators and the tangle insert operation.

So let us decompose G into a composition of trivalent tangles (“levels”), each of which has one “feature” and (possibly) some straight vertical strands. Note that by isotopy we can make sure that every level has strands ending at both its bottom and top, except for the first or the last level in the case of 1-tangles. An example of level decomposition is shown in the figure below. Note that the levels are generally not elements of sKTG (have too many ends). However, we can turn each of them into a (1, 1)-tangle (or a 1-tangle in case of the aforementioned top first or last levels) by “closing up” their tops and bottoms by arbitrary trees. In the example below we show this for one level of the Morse-drawn sKTG containing a crossing and two vertical strands.

1 2 3 4 5 6  3

Now we can compose the sKTG’s obtained from closing up each level. Each tree that we used to close up the tops and bottoms of levels determines a “parenthesization” of the strand endings. If these parenthesizations match on the top of each level with the bottom of the next, then we can recreate tangle composition of the levels by composing their closed versions followed by a number of unzips performed on the connecting trees. This is illustrated in the example below, for two consecutive levels of the sKTG of the previous example.

unzips^{3}^{4}  ^{3}^{4}

If the trees used to close up consecutive levels correspond to different parenthesizations, then we can use insertion of the left and right associators (the 5th and 6th pictures of the list of generators in the statement of the theorem) to change one parenthesization to match the other. This is illustrated in the figure below.

unzips

So far we have shown that G can be assembled from closed versions of the levels in its Morse drawing. The closed versions of the levels of G are simpler sKTG’s, and it remains to show that these can be obtained from the generators using sKTG operations.

=^{close}^{up}

Let us examine what each level might look like. First of all, in the absence of any “features” a level might be a single strand, in which case it is the first generator itself. Two parallel strands when closed up become the “bubble”, as shown on the right.

Now suppose that a level consists of n parallel strands, and that the trees used to close it up on the top and bottom are horizontal mirror images of each other, as shown below (if not, then this can be achieved by associator insertions and unzips). We want to show that this sKTG can be obtained from the generators using sKTG operations. Indeed, this can be achieved by repeatedly inserting bubbles into a bubble, as shown:

close^{up}  =

A level consisting of a single crossing becomes a left or right twist when closed up (depending on the sign of the crossing). Similarly, a single vertex becomes a bubble. A single minimum or maximum becomes a noose or a balloon, respectively.

It remains to see that the sKTG’s obtained when closing up simple features accompanied by more through strands can be built from the generators. A minimum accompanied by an extra strand gives rise to the sKTG obtained by sticking a noose onto a vertical strand (similarly, a balloon for a maximum). In the case of all the other simple features and for minima and maxima accompanied by more strands, we inserting the already generated elements into nested bubbles (bubbles inserted into bubbles), as in the example shown below. This completes the proof.

close^{up}  =
□

6. Glossary of notation

Greek letters, then Latin, then symbols:

δ  Satoh’s tube map  3.4
Δ  co-product  3.2
ι  inclusion 𝔱𝔯_{n} → P^{w}(↑_{n})  3.2
ν  the invariant of the unknot  4.7

π the projection P^{w}(↑_{n}) → 𝔞_{n} ⊕ 𝔱𝔡𝔢𝔯_{n} 3.2

ϕ  log of an associator  4.7
Φ  an associator  4.7
ψ_{β}  “operations”  2.1

𝔞_{n} n-dimensional Abelian Lie algebra 3.2 A a candidate associated graded structure

2.3 A^{sv} D^{v} mod 6T, RI 3.1 A^{sw} D^{w} mod \overrightarrow{4T}, TC, RI 3.1 A^{sw} grad wTF 4.3 A^{sw} grad wTF^{o} 4.6 A^{(s)w} A^{w} and/or A^{sw} 4.3 A^{u} chord diagrams mod rels for KTGs 4.7

A^{v}  D^{v} mod 6T  3.1
A^{w}  D^{w} mod \overrightarrow{4T}, TC  3.1
A^{w}  grad wTF^{o} without RI  4.3

A^{−}(↑_{n}) A^{−} for pure n-tangles 3.2 A_{e} 1D orientation reversal 4.2.3 Ass associative words 3.2 Ass^{+} non-empty associative words 3.2 B_{n}^{w} n-coloured unitrivalent arrow

diagrams 3.2

C  the invariant of a cap  4.4
CP  the Cap-Pull relation  4.2.2, 4.3

CW Cap-Wen relations 4.2.2

c  a chord in A^{u}  4.7
𝔡𝔢𝔯  Lie-algebra derivations  3.2

D^{v}, D^{w} arrow diagrams for v/w-tangles 3.1

div  the “divergence”  3.2
F  a map A^{w} → A^{w}  4.3
F  the main [AT] unknown  4.5
FR  Flip Relations  4.2.2, 4.3
fil  a filtered structure  2.3
I  augmentation ideal  2.2
J  a map TAut_{n} → exp(𝔱𝔯_{n})  3.2
j  a map TAut_{n} → 𝔱𝔯_{n}  3.2

KTG Knotted Trivalent Graphs 4.7

𝔩𝔦𝔢_{n}  free Lie algebra  3.2
l  a map 𝔱𝔡𝔢𝔯_{n} → P^{w}(↑_{n})  3.2
O  an “algebraic structure”  2.1
P_{n}^{w}  primitives of B_{n}^{w}  3.2

P^{−}(↑_{n}) primitives of A^{−}(↑_{n}) 3.2 grad associated graded structure 2.2 R the invariant of a crossing 4.4 R4 a Reidemeister move for

foams/graphs 4.2.2 𝔰𝔡𝔢𝔯 special derivations 3.3 S the circuit algebra of skeletons 2.4 SAut_{n} the group exp(𝔰𝔡𝔢𝔯_{n}) 4.7

S_{k}  complete orientation reversal  3.5
S_{e}  complete orientation reversal  4.2.3

sKTG signed long KTGs 4.7 TV Twisted Vertex relations 4.2.2 𝔱𝔡𝔢𝔯 tangential derivations 3.2

𝔱𝔯_{n}  cyclic words  3.2
𝔱𝔯_{n}^{s}  cyclic words mod degree 1  3.2

TAut_{n} the group exp(𝔱𝔡𝔢𝔯_{n}) 3.2

u  a map 𝔱𝔡𝔢𝔯_{n} → P^{w}(↑_{n})  3.2
u_{e}  strand unzips  4.2.3
uT  u-tangles  3.3

V, V^{+} the invariant of a (positive) vertex 4.4 V^{−} the invariant of a negative vertex 4.4

VI  Vertex Invariance  4.3
vT  v-tangles  3.1
W  Z(w)  4.4

W^{2} Wen squared 4.2.2

w  the wen  4.2.1
−  wenjugation  4.6
wT  w-tangles  3.1

wTF w-tangled foams with wens 4.1 wTF^{o} orientable w-tangled foams 4.6

Z  expansions  throughout
Z_{A}  an A-expansion  2.3
4T  4T relations  4.7
↑  a “long” strand  throughout
↑  the quandle operation  2.1
∗  the adjoint on A^{w}(↑_{n})  3.2

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Department of Mathematics, University of Toronto, Toronto Ontario M5S 2E4, Canada

  Email address: drorbn@math.toronto.edu
  URL: http://www.math.toronto.edu/~drorbn

School of Mathematics and Statistics, The University of Sydney, Carslaw Building, Camperdown, NSW, 2006, Australia

  Email address: zsuzsanna.dancso@sydney.edu.au