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FINITE TYPE INVARIANTS OF W-KNOTTED OBJECTS III: W-FOAMS, THE KASHIWARA-VERGNE THEOREM AND DRINFEL’D

ASSOCIATORS

DROR BAR-NATAN AND ZSUZSANNA DANCSO

Abstract. This is the third in a series of papers studying the finite type invariants of various w-knotted objects and their relationship to the Kashiwara-Vergne problem and Drinfel’d associators. In this paper we present a topological solution to the Kashiwara-Vergne problem. In particular we recover via a topological argument the Alkeseev-Enriquez-Torossian \underline{AlekseevEnriquezTorossian:ExplicitSolutions} [AET] formula for explicit solutions of the Kashiwara-Vergne equations in terms of associators.

We study a class of w-knotted objects: knottings of 2-dimensional foams and various associated features in four-dimensioanl space. We use a topological construction which we name the double tree construction to show that every expansion (also known as universal finite type invariant) of parenthesized braids extends first to an expansion of knotted trivalent graphs (a well known result), and then extends uniquely to an expansion of the w-knotted objects mentioned above.

In algebraic language, an expansion for parenthesized braids is uniquely determined by Drinfel’d associator Φ, and an expansion for w-knotted objects is uniquely determined by a solution V of the Kashiwara-Vergne problem \underline{KashiwaraVergne:Conjecture} [KV], as reformulated by Alekseev and

Torossian \underline{AlekseevTorossian:KashiwaraVergne} [AT]. Hence our result provides a topological framework for the result of \underline{AlekseevEnriquezTorossian:ExplicitSolutions} [AET]

that “there is a formula for V in terms of Φ”, along with an independent topological proof of the Kashiwara-Vergne Theorem and the Alekseev-Enriquez-Torossian formula.

Contents

1. Introduction  2
1.1. Executive Summary  2
1.2. Detailed Introduction  3
1.3. Computations  7
1.4. Paper Structure  7
2. The spaces w̃TF and A^{sw} in more detail  7
2.1. The generators of w̃TF  8
2.2. The relations  8

Date: first edition in future, this edition Aug. 3, 2023. The arXiv:????.???? edition may be older. 2020 Mathematics Subject Classification. 57M25. Key words and phrases. virtual knots, w-braids, w-knots, w-tangles, knotted graphs, finite type invari-

ants, Kashiwara-Vergne, associators, double tree, free Lie algebras.

The first author’s work was partially supported by NSERC grants RGPIN-264374 and RGPIN-2018-04350, and by the Chu Family Foundation (NYC), and wishes to thank the Sydney Mathematics Research Institute for their hospitality and support. The second author was partially supported by NSF grant no. 0932078 000 while in residence at the Mathematical Sciences Research Institute, and by the Australian Research Council DECRA DE170101128 Fellowship. Electronic version and related files at \underline{DoubleTree} [WKO3],

http://www.math.toronto.edu/~drorbn/papers/WKO3/.

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2.3. The operations  9
2.4. The associated graded structure A^{sw}  11
2.5. The homomorphic expansion  14
3. Proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1  16
3.1. Proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1 Part (1)  16
3.2. Proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1 Part (2)  18
3.3. Proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1 part (3): the double tree construction.  20
Appendix A. The Alekseev–Enriquez–Torossian formula  42
References  45

1. Introduction

1.1. Executive Summary. This brief section is a large-scale overview of the main result of this paper and the idea behind its proof; it is followed by a detailed introduction.

A homomorphic expansion for a class of topological objects K is an invariant Z: K → A whose target space A is canonically associated with K (its associated graded). Homomorphic expansions satisfy a certain universality property, and respect operations which exist on K and therefore also on A. Such invariants are often hard to find, and when they are found, they are often intimately connected with deep mathematics, in particular, quantum algebra and Lie theory:

• For many classes of knotted objects in 3-dimensional spaces homomorphic expansions don’t exist — for example, one would have loved ordinary tangles to have homomorphic expansions, but they don’t.

• Yet parenthesized tangles, or nearly-equivalently, knotted trivalent graphs in ℝ^{3} – which, for the purpose of this short summary, we denote by K^{u}, denoted sKTG later – do have homomorphic expansions. A homomorphic expansion Z^{u}: K^{u} → A^{u} is defined by its values on a couple of elements of K^{u} which generate K^{u} using the operations K^{u} is equipped with. The most interesting of these generators is the tetrahedron �, and Φ = Z^{u}(�) turns out to be equivalent to a Drinfel’d associator.

• A certain class of four-dimensional knotted graphs called w-foams – denoted K^{w} for now, wTF^{o} later in the paper – also has homomorphic expansions. The most interesting generator of K^{w} is the vertex �, and if Z^{w}: K^{w} → A^{w} is a homomorphic expansion, then it turns out that V = Z^{w}(�) is equivalent to a solution of the Kashiwara-Vergne problem in Lie theory.

u-ops  u-ops w-ops A^{u} Φ V A^{w} w-ops K^{w} � K^{u} �  Z^{u} ⊆ Z^{w}

Roughly speaking, K^{u} is a part of K^{w} and A^{u} is a part of A^{w}, as in the figure on the right. More precisely, there are natural maps a: K^{u} → K^{w} and α: A^{u} → A^{w}. The main purpose of this paper is to prove the following theorem, whose precise version is stated later

as Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1:

Theorem. Any homomorphic expansion Z^{u} for K^{u} extends uniquely to a homomorphic expansion Z^{w} for K^{w}, and therefore, any Drinfel’d associator Φ gives rise to a solution V of the Kashiwara-Vergne problem.

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The proof of this theorem is conceptually simple: we show that the generators of K^{w} can be explicitly expressed using the generators of K^{u} and the operations of K^{w}, and that the resulting explicit formulas for Z^{w}(�) (and for Z^{w} of the other generators) satisfie all the required relations.

The devil is in the details. It is in fact impossible to express the generators of K^{w} in terms of the generators of K^{u} — to do that, one first has to pass to a larger space K̃^{w} (in the paper w̃TF) that has more objects and more operations, and in which the desired explicit expressions do exist. But even in K̃^{w} these expressions are complicated, and in order to verify the relations they need to be expressed using the framework of a multi-step “double tree construction”. A brief pictorial summary of the construction is below, and the explanation takes up the bulk of this paper:

T  T  T  =  T

1.2. Detailed Introduction. This paper is the third in a sequence^{\underline{Bar-NatanDancso:WKO1, Bar-NatanDancso:WKO2, DoubleTree}} [WKO1, WKO2, WKO3] studying finite type invariants of w-knotted objects, and contains the strongest result: a topological construction for a homomorphic expansion of w-foams from the Kontsevich integral. This in particular implies the Kashiwara-Vergne Theorem of Lie theory, more precisely, it gives the ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET] formula for solutions of the Kashiwara-Vergne equations in terms of Drinfel’d associators.

The papers in this sequence need not be read consecutively. Readers broadly familiar with finite type invariants will have no trouble reading ^{\underline{Bar-NatanDancso:WKO2}} [WKO2] and this paper without having read ^{\underline{Bar-NatanDancso:WKO1}} [WKO1]. However, the setup and main results of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2] are used heavily in this paper. Reproducing all necessary details would be lengthy, but we include concise summaries for readers already familiar with the content, and otherwise refer to specific results or sections

of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2] throughout.

The Kashiwara-Vergne conjecture (KV for short) — proposed in 1978 ^{\underline{KashiwaraVergne:Conjecture}} [KV] and proven in 2006 by Alekseev and Meinrenken ^{\underline{AlekseevMeinrenken:KV}} [AM] — asserts that solutions exist for a certain set of equations in the space of “tangential automorphisms” of the free lie algebra on two generators. For a precise statement we refer the reader to ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.4] or ^{\underline{AlekseevTorossian:KashiwaraVergne}} [AT, Section 5.3]. The existence of such solutions has strong implications in Lie theory and harmonic analysis, in particular it implies the multiplicative property of the Duflo isomorphism, which

was shown to be knot-theoretic in ^{\underline{Bar-NatanLeThurston:TwoApplications, BDS:Duflo}} [BLT, BDS].

In ^{\underline{AlekseevTorossian:KashiwaraVergne}} [AT] Alekseev and Torossian gave another proof of the KV conjecture based on a deep connection with Drinfel’d associators. In turn, Drinfel’d’s theory of associators ^{\underline{Drinfeld:QuasiHopf}} [Dr] can be interpreted as a theory of well-behaved expansions (universal finite type invariants)

of parenthesized tangles^{1} ^{\underline{LeMurakami:Universal, Bar-Natan:NAT}} [LM, BN2], or of knotted trivalent graphs ^{\underline{Dancso:KIforKTG}} [Da]. In ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET] Alek-

seev, Enriquez and Torossian gave an explicit formula for solutions of the Kashiwara-Vergne equations in terms of Drinfel’d associators.

1“q-tangles” in \underline{LeMurakami:Universal} [LM], “non-associative tangles” in \underline{Bar-Natan:NAT} [BN2].

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Figure 1. Two examples of parenthesized braids. Note that by convention the parenthetization can be read from the distance scales between the endpoints of the braid, and so we omit

the parentheses in parts of this paper.  fig:PBexample

In ^{\underline{Bar-NatanDancso:WKO2}} [WKO2] we re-interpreted the Kashiwara-Vergne conjecture as the problem of finding a “homomorphic” expansion of a class of knotted trivalent tubes – w-foams or w-tangled foams – in 4-dimansional space, and explained the connection to Drinfel’d associators in terms of a relationship between 3-dimensional and 4-dimensional topology. Another topological interpretation for the KV problem in terms of the Goldman-Turaev Lie bialgabra later emerged in ^{\underline{AKKN:GoldmanTuraev, AKKN:GTReverse}} [AKKN1, AKKN2], and the papers ^{\underline{Massuyeau:GT}} [M] and ^{\underline{AlekseevNaef:GTKZ}} [AN] contain constructions of Goldman-Turaev expansions from the Kontsevich integral and the Knizhnik-Zamolodchikov connection, respectively.

The algebraic structure of w-foams is built on the concept of a circuit algebra, which was also identified as equivalent to the operadic structure of a wheeled prop in ^{\underline{DHR:CircAlg}} [DHR1]. The symmetry groups of Kashiwara-Vergne solutions, called the Kashiwara-Vergne groups, are shown to be automorphism groups of the w-foam circuit algebra and its associated graded arrow diagrams in ^{\underline{DHR:KVKRV}} [DHR2]. The relationship between the symmetries of Drinfel’d associators

– the Grothendieck-Teichmuller groups – and the Kashiwara-Vergne groups is described in the topological context of w-foams in ^{\underline{DHaR:GRTKRV}} [DHoR].

In this paper we present a topological construction for a homomorphic universal finite type invariant of w-foams, thereby giving a new topological proof for the KV conjecture. This construction also leads to an explicit formula for KV-solutions in terms of Drinfel’d associators, which we prove agrees with the formula ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET, Theorem 4]. Computational verification of these results is given in ^{\underline{Bar-Natan:WKO4}} [WKO4]. We conclude this introduction with a brief summary of the topological and algebraic tools developed in this paper, and state the main result.

1.2.1. Topology. We begin by describing a chain of maps from “parenthesized braids” to “(signed) knotted trivalent graphs” to “w-tangled foams”:

K := {uPaB ⟶^{cl} sKTG ⟶^{a} w̃TF }.

Parenthesized braids are braids whose ends are ordered along two lines, the “bottom” and the “top”, along with parenthetizations of the endpoints on the bottom and on the top. Two examples are shown in Figure ^{\underline{fig:PBexample}}^{\underline{fig:PBexample}} 1. Parentehesized braids form a category whose objects are parenthetizations, morphisms are the parenthesized braids, and composition is given by stacking. In addition to stacking, there are several operations defined on parenthesized braids: strand addition, removal and doubling. A detailed introduction to parenthesized braids is in ^{\underline{Bar-Natan:GT1}} [BN1].

DRAFT

− + + + −

Trivalent graphs are oriented graphs with three edges meeting at each vertex and whose vertices are equipped with a cyclic orientation of the incident edges. A knotted trivalent graph (KTG) is a framed embedding of a trivalent graph into ℝ^{3}. KTGs are studied from a finite type invariant point of view in ^{\underline{Bar-NatanDancso:KTG}} [BND1]. In this paper we use a version of KTGs that was introduced and studied in \underline{Bar-NatanDancso:WKO2} [WKO2, Section 4.6], namely trivalent tangles with one or two ends and with some extra combinatorial information: trivalent vertices are equipped with a marked “distinguished edge” and signs. We call this space sKTG (for signed KTGs), as in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2]. An example is shown on the right. The space sKTG is also equipped with several operations: tangle insertion, sticking a 1-tangle onto an edge of another tangle, disjoint union of 1-tangles, edge unzip, and edge orientation switch (see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.6] for details).

The space w̃TF is a minor extension of the space wTF^{o} studied in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.1 – 4.4], and will be introduced in detail in Section ^{\underline{sec:wTFe}}^{\underline{sec:wTFe}} 2. Algebraically, it is described as a circuit algebra: similar to a planar algebra but with non-planar connections allowed, see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 2.4] for a concise introduction, or ^{\underline{DHR:CircAlg}} [DHR1, Section 2] for a rigorous discussion. As a circuit algebra, w̃TF is generated by various kinds of crossings and “vertices”, as well as “caps”, modulo certain relations (Reidemeister moves), and equipped with a number of auxiliary operations beyond the circuit algebra compositions. This Reidemeister theory locally – and conjecturally globally – represents knotted tubes in 4-dimensional space with singular foam vertices, caps, and attached one-dimensional strings.

The map cl : uPaB → sKTG is the “closure map”. Given a parenthesized braid, close up its top and bottom each by gluing a binary tree according to the parentetization; this produces a sKTG with the convention that all strands are oriented upwards, bottom vertices are negative, and top vertices are positive. An example is shown below.

cl  _{−} _{−}^{+}  ^{+}^{+} _{−}  (1) eq:cl

The map a : sKTG → w̃TF arises combinatorially from the fact that all sKTG diagrams can be interpreted as elements of w̃TF, and sKTG Reidemeister moves are also imposed in w̃TF. Topologically, it is an extended version of Satoh’s tubing map, described in Remark 3.1.1 of

\underline{Bar-NatanDancso:WKO2} [WKO2].

1.2.2. Algebra. The chain of maps K is an example of a general “algebraic structure”, as discussed in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 2.1]. An algebraic structure consists of a collection of objects belonging to a number of “spaces” or “different kinds”, and operations that may be unary, binary, multinary or nullary, between these spaces. In this case there are many spaces (or kinds of objects): for example, parenthesized braids with specified bottom and top parenthetizations form one space, so do knottings of a given trivalent graph (skeleton). There is a large collection of operations, consisting of all the internal operations of uPaB, sKTG and w̃TF, as well as the maps a and cl.

DRAFT

In Sections 2.1 to 2.3 of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2]we discuss associated graded structures and expansions for general algebraic structures. For any algebraic structure (think braids, or tangles with tangle composition), one allows formal linear compositions of elements of the same kind (in this case, knotted objects with the same skeleton: braids with the same underlying permutation, or tangles with the same connectivity of endpoints and same number of circle components).

Associated graded structures are taken with respect to the filtration by powers of the augmentation ideal. For the spaces uPaB, sKTG and w̃TF, the associated graded spaces A^{hor}, A^{u} and A^{sw} are the spaces of “horizontal chord diagrams on parenthesized strands”, “chord diagrams on trivalent skeleta”, and “arrow diagrams”, as described in ^{\underline{Bar-Natan:GT1}} [BN1], ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.6], and Section ^{\underline{sec:wTFe}}^{\underline{sec:wTFe}} 2 of this paper, respectively. Thus, the associated graded structure of K is

A := {A^{hor} ⟶^{cl} A^{u} ⟶^{α} A^{sw}},

where cl and α are the maps induced by cl and a, respectively. More specifically, cl is the “closure of chord diagrams”, and α is “replacing each chord with the sum of its two possible orientations”, see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 3.3].

An expansion ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 2.3] is a filtered map from an algebraic structure to its associated graded structure, whose associated graded map is the identity. In knot theory, expansions are also known as universal finite type invariants. A homomorphic expansion is an expansion which behaves well with respect to the operations of the algebraic structure, that is, it intertwines each operation with its induced counterpart on the associated graded structure. (For a more detailed introduction see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 2.3].) Hence, a homomorphic expansion Z : K → A is a triple of homomorphic expansions Z^{b}, Z^{u}, and Z^{w} for K^{b} := uPaB, K^{u} := sKTG and K^{w} := w̃TF, respectively, so that the following diagram commutes:

\xymatrix{K : \ar[d]^{Z} & K^{b} \ar[r]^{cl} \ar[d]^{Z^{b}} & K^{u} \ar[r]^{a} \ar[d]^{Z^{u}} & K^{w} \ar[d]^{Z^{w}} \\ A : & A^{hor} \ar[r]^{cl} & A^{u} \ar[r]^{α} & A^{w}}  (2) eq:MainDiag

1.2.3. The main result. We recall ^{\underline{Bar-Natan:GT1}} [BN1] that a homomorphic expansion Z^{b} for parenthesized braids is determined by a horizontal chord associator or Drinfel’d associator Φ = Z^{b}( ). A homomorphic expansion Z^{u} of sKTG is also determined^{2} by a Drinfel’d associator (horizontal chords or not; see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.6]), so the significance of the left commutative square is to force the associator corresponding to Z^{u} to be a horizontal chord associator. In turn, Z^{w} is determined by a solution F to the Kashiwara-Vergne problem (see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.4 – 4.5]). The goal of this paper is to prove the following theorem, which, via the correspondence above, implies the Kashiwara-Vergne Theorem:

thm:main Theorem 1.1. (1) Assuming that Z : K → A exists, it is determined^{3} by Z^{u}.

(2) Let C = Z^{w}(�), then C is given by a power series with even part C_{0} = α(ν^{1/4}), where ν is the Kontsevich integral of the unknot. There is a formula for V = Z^{w}( )) in

2With the exception of some minor normalization, see \underline{Bar-NatanDancso:WKO2} [WKO2], Lemma 4.14 and the paragraph after.

3In fact, almost entirely determined by Z^{b}, with the exception of some minor normalization of Z^{u} which is not determined by an associator.

DRAFT

terms of C and the Drinfel’d associator Φ associated to Z^{u}:

V = C_{1}^{−1}C_{2}^{−1}φ (Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}) · e^{a_{23}/2}Φ(a_{23}, a_{43})) C_{(12)},  (3) eqn:AET

where a denotes a single arrow^{4}. This agrees^{5} with the formula proven in ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET].

(3) Every Z^{b} extends to a Z.

Remark 1.2. The formula in part (2) of the Theorem, expresses V = Z^{w}( )) in terms of the Drinfel’d associator Φ, and C = Z^{w}(�). One might wonder if there are separate formulas for both V and C in terms of Φ. In fact, in Corollary ^{\underline{cor:CapValue}}^{\underline{cor:CapValue}} 3.13 we compute the “even part” of C explicitly, and show that it is fixed, in other words, does not depend on Φ. Part (1) of the Theorem, proven in Section ^{\underline{subsec:Part1Proof}}^{\underline{subsec:Part1Proof}} 3.1.2, shows that the complete value of C (even and odd part) is uniquely determined by Φ. In Theorem ^{\underline{thm:Unitarity}}^{\underline{thm:Unitarity}} 3.6 we also prove that a suitable value exists for any choice Φ.

The key to the proof of the theorem is to show that the generator of w̃TF can be expressed in terms of the generator of uPaB and the operations of K. Assuming that Z exists, this yields a formula for V in terms of Φ.

1.3. Computations. We note that this paper is “abstract”, yet everything difficult in it occurs within graded spaces, and can be computed explicitly up to a certain degree. The highest degree to which computations can be completed depends on the specific object. In a follow-up paper ^{\underline{Bar-Natan:WKO4}} [WKO4] many of these computations are carried out.

1.4. Paper Structure. In Section ^{\underline{sec:wTFe}}^{\underline{sec:wTFe}} 2 we provide an overview of the space wTF^{o} of (oriented) w-foams and its extension with strings w̃TF. We provide a brief review of definitions and crucial facts from ^{\underline{Bar-NatanDancso:WKO2}} [WKO2], and details of the extension. We prove that homomorphic expansions for wTF^{o} extend uniquely to homomorphic expansions for w̃TF.

Section ^{\underline{sec:Proof}}^{\underline{sec:Proof}} 3 makes up the bulk of the paper and is devoted to the proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1. In Section ^{\underline{subsec:Part1}}^{\underline{subsec:Part1}} 3.1 we prove part (1). In Section ^{\underline{subsec:AETFormula}}^{\underline{subsec:AETFormula}} 3.2 we deduce the formula for Kashiwara-Vergne solutions in terms of Drinfel’d associators, proving part (2). In Section ^{\underline{subsec:DT}}^{\underline{subsec:DT}} 3.3 we prove statement (3), the hardest part of the proof.

Section ^{\underline{sec:Rmks}}^{\underline{sec:Rmks}} ?? is a short section of closing remarks, and in Appendix ^{\underline{app:AET}}^{\underline{app:AET}} A we give an explicit comparison and equivalence between our formula in Part (2) and the ALekseev–Enriquez–

Torossian ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET] formula.

2. The spaces w̃TF and A^{sw} in more detail

sec:wTFe

As mentioned in the introduction, w̃TF is a minor extension of the space wTF^{o} studied in \underline{Bar-NatanDancso:WKO2} [WKO2, Section 4.1 – 4.4]. It can be introduced as a planar algebra or as a circuit algebra; we will do the latter as it is simpler and more concise. Circuit algebras are defined in \underline{Bar-NatanDancso:WKO2} [WKO2, Section 2.4]; in short, they are similar to planar algebras but without the planarity requirement for “connecting strands”. As in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2], each generator and relation of w̃TF has a local topological interpretation. Recall from ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Sections 1.2, 3.4, 4.1] that wTF^{o} diagrams represent certain ribbon knotted tubes with foam vertices in ℝ^{4}, and the circuit algebra wTF^{o} is conjecturally a Reidemeister theory for this space (i.e., there is a surjection δ

4The notation is explained in detail in Section \underline{subsec:AETFormula} \underline{subsec:AETFormula} 3.2

5The two formulas are written in different languages, and checking that they agree takes effort. See Section \underline{subsec:AETFormula} \underline{subsec:AETFormula} 3.2 and Appendix \underline{app:AET} \underline{app:AET} A.

DRAFT

from the circuit algebra wTF^{o} to ribbon knotted tubes with foam vertices, and δ is conjectured to be an isomorphism). The space w̃TF extends wTF^{o} by adding one-dimensional strands to the picture. Note that in themselves, one dimensional strands in ℝ^{4} are never knotted, however, they can be knotted with the two-dimensional tubes. In figures two-dimensional tubes will be denoted by thick lines and one dimensional strings by thin red lines. With this in mind, we define w̃TF as a circuit algebra defined in terms of generators and relations, and with some extra operations beyond circuit algebra compositions. Each generator, relation and operation has a local topological interpretation which provides much of the intuition behind the proofs. However, the corresponding Reidemeister theorem is only conjectural.

w̃TF = CA⟨ , ,  ,  , , , , , generators  | relations as in Section ^{\underline{subsec:wrels}}^{\underline{subsec:wrels}} 2.2 | auxiliary operations as in Section ^{\underline{subsec:wops}}^{\underline{subsec:wops}} 2.3 ⟩

subsec:wgens

2.1. The generators of w̃TF. We begin by discussing the local topological meaning of each generator shown above.

The first five generators are as described in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Sections 4.1.1], we briefly recall their descriptions here. Knotted (more precisely, braided) tubes in ℝ^{4} can equivalently be thought of as movies of flying rings in ℝ^{3}. The two crossings stand for movies where two rings trade places by the ring of the under strand flying through the ring of the over strand. The dotted end represents a tube “capped off” at the bottom by a disk. Generators 4 and 5 stand for singular “foam vertices”, and will be referred to as the positive and negative vertex, respectively. The positive vertex represents the movie shown on the left: the right ring approaches the left ring from below, flies inside it and merges with it. The negative vertex represents a ring splitting and the inner ring flying out below and to the right. To be completely precise, w̃TF as a circuit algebra has more vertex generators than

shown above: the vertices appear with all possible orientations of the strands. However, all other versions can be obtained from the ones shown above using “orientation switch” operations (to be discussed in Section ^{\underline{subsec:wops}}^{\underline{subsec:wops}} 2.3).

The thin red strands denote one dimensional strings in ℝ^{4}, or “flying points in ℝ^{3}”. The crossings between the two types of strands (generators 6 and

7) represent “points flying through rings”. For example, the picture on the left shows generator 6, where “the point on the right approaches the ring on the left from below, flies through the ring and out to the left above it”. This explains why there are no generators with a thick strand crossing under a thin red strand: a ring cannot fly through a point.

Generator 9 is a trivalent vertex of 1-dimensional strings in ℝ^{4}. Finally, the last generator is a mixed vertex: a one-dimensional string attached to the wall of a 2-dimensional tube, as shown in Figure ^{\underline{fig:MixedVertex}}^{\underline{fig:MixedVertex}} 2. All generators should be shown in all possible strand orientation combinations; we are suppressing this to save space.

subsec:wrels

2.2. The relations. As a list, the relations for w̃TF are the same as the relations for wTF^{o} \underline{Bar-NatanDancso:WKO2} [WKO2, Section 4.5]: \{R1^{s}, R2, R3, R4, OC, CP\}. Recall that R1^{s} is the weak (framed) version of the Reidemeister 1 move; R2 and R3 are the usual second and third Reidemeister

DRAFT

Figure 2. A string-tube vertex.  fig:MixedVertex

OC:  CP:
Figure 3. The OC and CP relations.  fig:wTFeRels

moves; R4 allowes moving a strand over or under a vertex. OC stands for Overcorssings Commute, CP for Cap Pullout: these two relations are shown in Figure ^{\underline{fig:wTFeRels}}^{\underline{fig:wTFeRels}} 3, for a detailed explanation see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.1.2].

In w̃TF all relations should be interpreted in all possible combinations of strand types and orientations (tube or string), for example the lower strand of the R2 relation can either be thick black or thin red, as shown below:

R2:

Similarly, any of the lower strands of the R3, R4, and OC relations may be thin red.

As in wTF^{o}, the relations all have local topological meaning and conjecturally w̃TF is a Reidemeister theory for ribbon knotted tubes in ℝ^{4} with caps, singular foam vertices and attached strings. For example, Reidemeister 2 with a thin red bottom strand is imposed because the movie where a point flies in through a ring and then immediately flies back out is isotopic to the movie where there is no interaction between the point and ring at all.

It is easy to verify that all relations represent local isotopies of welded (ribbon knotted) tubes in ℝ^{4} with singular vertices and attached strings. What is not clear at this stage is that this is a complete Reidemeister theory, that is, whether this is a complete set of relations. For more detail on this see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 1.2].

subsec:wops

2.3. The operations. Like wTF^{o}, w̃TF is equipped with a set of auxiliary operations in addition to the circuit algebra structure.

The first of these is orientation reversal. For the thin (red) strands, this simply means reversing the direction of the strand. For the thick strands (tubes), orientation switch comes in two versions. Recall from ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 3.4] that in the topological interpretation of wTF^{o}, each tube is oriented as a 2-dimensional surface, and also has a distinguished “core”:

DRAFT

\begin{aligned}& u_{e} \\ e \end{aligned}  ^{e  u_{e}}
Figure 4. Unzip and disc unzip.  fig:DiscUnzip

a line along the tube which is oriented as a 1-dimensional manifold and determines the “direction” or “1-dimensional orientation” of the tube. Both of these are determined by the direction of the strand in the circuit algebra, via Satoh’s tubing map.

Topologically, the operation “orientation switch”, denoted S_{e} for a given strand e, acts by reversing both the (1-dimensional) direction and the (2-dimensional) orientation of the tube e. Diagrammatically, this corresponds to simply reversing the direction of the corresponding strand e.

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

The “adjoint” operation, denoted A_{e}, on the other hand only reverses the (1-dimensional) direction of the tube e, not the orientation as a surface. Diagrammatically, this manifests itself as reversing the strand direction and adding two virtual crossings on either side of each crossing where e crosses over another strand, as shown on the right (note that the strand below e may be thick or thin). Note that virtual crossings don’t appear when e crosses under another strand. For more details on orientations and orientation switches, see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Sections 3.4 and 4.1.3].

The unzip operation u_{e} doubles the strand e using the blackboard framing, and then attaches the ends of the doubled strand to the connecting ones, as shown in Figure ^{\underline{fig:DiscUnzip}}^{\underline{fig:DiscUnzip}} 4. We restrict unzip to strands whose two ending vertices are of different signs. (For the definition of crossing and vertex signs, see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Sections 3.4 and 4.1].) Topologically, the blackboard framing of the diagram induces a framing of the corresponding tube in ℝ^{4} via Satoh’s tubing map, and unzip is the act of “pushing the tube off of itself slightly in the framing direction”. Note that unzips preserve the ribbon property.

A related operation, disc unzip, is unzip done on a capped strand, pushing the tube off in the direction of the framing (in diagram world, in the direction of the blackboard framing), as before. An example is shown in Figure ^{\underline{fig:DiscUnzip}}^{\underline{fig:DiscUnzip}} 4; see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.1.3] for details on framings and unzips.

So far all the operations we have introduced had already existed in wTF^{o}. There is also a new operation is called “puncture”, denoted p_{e}, which diagrammatically simply turns the thick black strand e into a thin red one. The corresponding topological picture is “puncturing a tube”, i.e., removing a small disk from it and retracting the rest to its core. Any crossings where e passes under another strand are not affected, while crossings in which e is the over strand turn into virtual crossings.

For simplicity, we place a restriction on which strands can be punctured, namely at each (fully thick black) vertex puctures are only allowed for one of the three meeting strands, as shown on the left of Figure ^{\underline{fig:punctures}}^{\underline{fig:punctures}} 5. More general puctures could be allowed in a theory with more than one kind of “string to tube” vertex. The right of the same figure shows that when

DRAFT

e  p_{e}  e p_{e}

Figure 5. Puncture operations: the picture on the left shows which edges can be punctured

at each vertex. The middle and right pictures show the effect of puncture operations.  fig:punctures

+ = +  \overrightarrow{4T} =

Figure 6. The TC and \overrightarrow{4T} relations. Note that the 3rd strand in each term of the \overrightarrow{4T} relation

can be either thick black or thin red, the relation applies in either case.  fig:TCand4T

puncturing one of the thick strands of a mixed vertex, the puncture “spreads”. Topologically, this is because the mixed vertex represents a string attached to a tube, so when puncturing e, the entire tube retracts to its core. Finally, a capped tube disappears (deformation retracts to a point) when punctured.

In summary,

w̃TF = CA ⟨ , ,  ,  , , , , , generators  | R1^{s}, R2, R3, ; R4, OC, CP ; relations| S_{e}, A_{e}, ; u_{e}, d_{e}, p_{e} ; auxiliary ; operations ⟩

2.4. The associated graded structure A^{sw}. As in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2], the space w̃TF is filtered by powers of the augmentation ideal and its associated graded circuit algebra, denoted A^{sw}, is a “space of arrow diagrams on foam skeletons with strings”. As a circuit algebra, A^{sw} is presented as follows:

A^{sw} = CA ⟨  , , , , , ,  generators  | relations as below | auxiliary operations as below ⟩ .

Generators 1 and 5 are called single arrows and they are of degree one, while all others are “skeleton features” of degree zero. The relations are almost the same as in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.2.1], which describes the relations for the associated graded of wTF^{o}: \overrightarrow{4T} (the 4-Term relation), TC (Tails Commute), RI (Rotation Invariance), CP (the arrow Cap Pullout), and VI (Vertex Invariance). For w̃TF there is an additional relation TF (Tails Forbidden on strings). The TC and \overrightarrow{4T} relations are shown in Figure ^{\underline{fig:TCand4T}}^{\underline{fig:TCand4T}} 6. The Vertex Invariance relation is shown in Figure ^{\underline{fig:VI}}^{\underline{fig:VI}} 7: here the ± signs depend on the strand orientations. Note that the type of the vertex and the types of each strand (thick black or thin red) are left undetermined: the VI relation applies in all cases. Figure ^{\underline{fig:RICPTF}}^{\underline{fig:RICPTF}} 8 shows the other relations: RI, CP and TF. Note that technically TF is not a relation: there were no generators with an arrow tail on a thin red strand, so saying that such an element vanishes is meaningless. However, without TF the VI relation would have to be stated for all the sub-cases of 0, 1 or 3 thin red strands, so we prefer this cleaner way, even if it is a slight abuse of notation.

DRAFT

±  ±  ±  = 0,  and ±  ±  ±  = 0.

Figure 7. The VI relation: the vertices and strands could be of any type, but the same

throughout the relation.  fig:VI

= RI  CP  TF = 0  = 0
Figure 8. The RI and CP relations, and the TF relation (which is not really a relation).  fig:RICPTF

Denote arrow diagrams on a given skeleton S by A^{sw}(S). In particular, A^{sw}(↑_{n}) denotes arrow diagrams on n (black) vertical strands, and A(�_{n}) denotes arrow diagrams on n capped strands.

Each operation on w̃TF induces a corresponding operation on A^{sw}. Orientation switch, adjoint, unzip, cap unzip, and long strand deletion act exactly the same way as they do for wTF^{o}^{o}. We quickly recall these here, for details see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.2.2]. The orientation switch S_{e} reverses the orientation of the skeleton strand e, and multiplies the arrow diagram by (−1)^{#{arrow heads and tails on e}}. The adjoint operation also reverses the skeleton strand e and multiplies the arrow diagram by (−1)^{#{arrow heads on e}}. Given a skeleton S with a distinguished strand e, unzip (or disc unzip, if e is capped) is an operation u_{e} : A^{sw}(S) → A^{sw}(u_{e}(S)) which maps each arrow ending on e to a sum of two arrows, one ending on each of the two new strands which replace e. Deleting a long strand e kills all arrow diagrams with any arrow ending on e. The operation induced by puncture, denoted p_{e}, turns the formerly thick black e into a thin red strand, and kills any arrow diagram with any arrow tails on e.

To summarise:

A^{sw} = CA ⟨  , , , , , ,  generators  | \overrightarrow{4T}, TC, VI, ; CP, RI, TF ; relations | S_{e}, A_{e}, u_{e}, d_{e}, p_{e} auxiliary operations⟩

As in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Definition 3.7], we define a “w-Jacobi diagram” (or just “arrow diagram”) by also allowing trivalent chord vertices, each of which is equipped with a cyclic orientation, and modulo the \overrightarrow{ST U} relations of Figure ^{\underline{fig:ASIHXSTU}}^{\underline{fig:ASIHXSTU}} 9. Denote the circuit algebra of formal linear combinations of these w-Jacobi diagrams by A^{swt}. Then, as in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Theorem 3.8], we have the following “bracket-rise” theorem:

Theorem 2.1. The natural inclusion of diagrams induces a circuit algebra isomorphism A^{sw} ≅ A^{swt}. Furthermore, the \overrightarrow{AS} and \overrightarrow{IHX} relations of Figure ^{\underline{fig:ASIHXSTU}}^{\underline{fig:ASIHXSTU}} 9 hold in A^{swt}.

The proof is identical to the proof of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Theorem 3.8]. In light of this isomorphism, we will drop the extra “t” from the notation and use A^{sw} to denote either of these spaces.

DRAFT

+  = 0 \overrightarrow{AS}  =  − \overrightarrow{IHX} =^{\overrightarrow{ST U}_{2}} =  −^{\overrightarrow{ST U}_{1}}  −  =^{\overrightarrow{ST U}_{3}}

Figure 9. The \overrightarrow{AS}, \overrightarrow{IHX} and the three \overrightarrow{ST U} rerations. Note that in \overrightarrow{ST U}_{1}, the skeleton

strand can be thin red or thick black, and that \overrightarrow{ST U}_{3} is the same as the TC relation.  fig:ASIHXSTU

Figure 10. An example of a tree, left, and a wheel, right.  fig:TreeAndWheel

The space A^{sw}(↑_{n}) forms a Hopf algebra with the stacking product and the standard co-product. (The coproduct is the sum over all possible ways of distributing the connected components of the arrow graph between two copies of the skeleton.) As in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2], the primitive elements of A^{sw}(↑_{n}) are connected diagrams, denoted P^{sw}(↑_{n}), and P^{sw}(↑_{n}) = ⟨trees⟩ ⊕ ⟨wheels⟩ as a vector space. Examples of trees and wheels are shown in Figure ^{\underline{fig:TreeAndWheel}}^{\underline{fig:TreeAndWheel}} 10; for details see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 3.1]. Note that the RI relation can now be rephrased (via \overrightarrow{ST U}_{2}) as the vanishing of the wheel with a single spoke, or one-wheel.

We recall the following crucial facts ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Proposition 3.19, Lemmas 4.6 and 4.7]:

fact:semidirect Fact 2.2. As a Lie algebra, P^{sw}(↑_{n}) ≅ ⟨wheels⟩ ⋊ ⟨trees⟩. The vector space (abelian Lie algebra) spanned by wheels is canonically isomorphic to the space (cyc_{n})_{≥1} of cyclic words^{6} in n letters of degree at least 2, where degree is given by word length, and degree 1 is killed by the RI relation.

fact:CapIsWheels Fact 2.3. A^{sw}(�), the part of A^{sw} with skeleton a single capped strand, is isomorphic as a vector space to the completed polynomial algebra freely generated by wheels w_{k} with k ≥ 2.

fact:VTwoStrands Fact 2.4. A^{sw}( ) ≅ A^{sw}(↑_{2}), where A^{sw}( ) stands for the space of arrow diagrams whose skeleton is a single vertex (the picture shows a positive vertex but the statement is true for all kinds of vertices with thick black strands), and A^{sw}(↑_{2}) is the space of arrow diagrams on two (thick black) strands.

The following Lemma – called the Sorting Lemma as we will see it “sorts” arrow tails above arrow heads – will play an important role. In particular the second isomorphism stated is the map φ appearing in Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1, part (2). We will refer to the isomorphism φ in the Lemma as the sorting isomorphism.

6Cyclic words are denoted 𝔱𝔯_{n} in \underline{Bar-NatanDancso:WKO2} [WKO2] and \underline{AlekseevTorossian:KashiwaraVergne} [AT] and 𝔗_{n} in \underline{AlekseevEnriquezTorossian:ExplicitSolutions} [AET].

DRAFT

x ψ x  h tx x  φ h  t
Figure 11. Inverse maps.  fig:SlideUpLemma

lem:CapString Lemma 2.5 (Sorting Lemma). There is a linear isomorphism φ : A^{sw}( ) ⟶^{≅} A^{sw}(↑) between the vector spaces of arrow diagrams on the indicated skeleta. On the left, the thin red string is a tangle end. The black strand may continue past the arrow, and there may be additional skeleton components: the same on both sides. Applying the isomorphism φ twice, one obtains A^{sw}( ) ≅^{φ} A^{sw}(↑_{2}).

Proof. We construct inverse maps between the two spaces. There is a natural map A^{sw}(↑) →^{ψ} A^{sw}( ), shown in Figure ^{\underline{fig:SlideUpLemma}}^{\underline{fig:SlideUpLemma}} 11: given an arrow diagram on a single thick black strand, place all arrow endings (denoted “x”) on the strand above the tube/string vertex.

In the other direction, consider an arrow diagram on the capped/stringed vertex. One may assume that there are only arrow tails on the capped strand under the vertex: any arrow head may be commuted using \overrightarrow{ST U} relations towards the cap, where it is killed by the CP relation^{7}. On the thin red strand there are only arrow heads. To construct φ, first “push” the arrow tails (denoted “t”) from the capped strand up across the vertex using the VI relation. Since tails vanish on the thin red strand, they simply slide past the vertex. Once the capped side is cleared, continue by sliding the arrow heads “h” up from the thin red string to the strand above the vertex. Now the cap relation kills any arrow heads on the capped strand, so once again they simply slide past the vertex. The result placed on a single thick black strand is shown in Figure ^{\underline{fig:SlideUpLemma}}^{\underline{fig:SlideUpLemma}} 11.

It is clear that ψ is well-defined, we leave it to the reader to check that so is φ as a short exercise. Given that both maps are well-defined, it is clear that they are inverses of each other. □

Observe that in the image of φ, all arrow tails are above arrow heads along the strand. Arrow diagrams of this form appear in the context of “over-then-under” tangles, which have applications in several contexts, including virtual braid classification ^{\underline{BDV:OU}} [BDV].

2.5. The homomorphic expansion. As discussed in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 2.3], an expansion for w̃TF is a map Z^{w} : w̃TF → A^{sw} with the property that the associated graded map gr Z^{w} : A^{sw} → A^{sw} is the identity map id_{A^{sw}}. A homomorphic expansion is an expansion which also intertwines each operation of w̃TF with its arrow diagrammatic counterpart. In \underline{Bar-NatanDancso:WKO2} [WKO2, Theorems 4.9 and 4.11] we proved that the existence of solutions for the Kashiwara– Vergne equations implies that there exsists a homomorphic expansion for wTF^{o}. In fact that

7This argument also appears in \underline{Bar-NatanDancso:WKO2} [WKO2], for example as the basic idea for the proof of Fact \underline{fact:CapIsWheels} \underline{fact:CapIsWheels} 2.3.

DRAFT

homomorphic expansions^{8} for wTF^{o} are in one-to-one correspondence with solutions to the Kahiwara-Vergne problem.

The point of this paper is to provide a topological construction for such a homomorphic expansion (and hence for a solution of the Kashiwara–Vergne conjecture), and this is easier to do for the slightly more general space w̃TF.

Let A^{osw} ⊆ A^{sw} denote arrow diagrams on wTF^{o} skeleta, the associated graded space of wTF^{o}. One of the key results of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.3] is the characterisation of homomorphic expansions of wTF^{o}. For any (group-like) homomorphic expansion Z^{ow} : wTF^{o} → A^{osw}, the value Z^{ow}(�) is uniquely determined and equals R = e^{a_{12}}, where a_{12} denotes a single arrow from the over strand 1 to the under strand 2.

To state the full characterisation, we use co-simplicial notation in subscripts. For example, for R = e^{a_{12}} =∈ A^{sw}(↑_{2}), R_{13} = e^{a_{13}} and R_{23} = e^{a_{23}} in A^{sw}(↑_{3}) are the diagrams where R is placed on strands 1 and 3, and 2 and 3, respectively. R_{(12)3} ∈ A^{sw}(↑_{3}) is obtained by doubling the first strand of R and placing it on strands 1 and 2, and placing the second strand of R on strand 3, that is, R_{(12)3} = e^{a_{13}+a_{23}}. Similarly for V ∈ A(↑_{2}), V_{12} ∈ A(↑_{3}) denotes V placed on the first two starnds, et cetera.

fact:EquationsForZ Fact 2.6. A filtered, group-like map Z^{ow} : wTF^{o} → A^{osw} is a homomorphic expansion if and

only if the Z^{ow}-values V = Z^{ow}( ) and C = Z^{ow}(�) satisfy the following equations:

(1) R4 Equation:

V_{12}R_{(12)3} = R_{23}R_{13}V_{12} in A^{sw}(↑_{3}).  (R4) eq:R4

(2) Unitarity Equation:

V · A_{1}A_{2}(V) = 1 in A^{sw}(↑_{2}),  (U) eq:U

where A_{1} and A_{2} denote the antipode operations.

(3) Cap Equation^{9}:

C_{(12)}V_{12}^{−1} = C_{1}C_{2} in A^{sw}(�_{2}),  (C) eq:C

where the subscripts mean strand placements as in the R4 Equation.

We begin by showing that finding a homomorphic expansion for w̃TF is no harder than finding one for wTF^{o}.

thm:ExtendRestrict Theorem 2.7. Homomorphic expansions for wTF^{o} are in one-to-one correspondence with homomorphic expansions for w̃TF via unique extension and restriction.

\xymatrix{wTF^{o} \ar@{^{(}->}[r] \ar[d]^{Z^{ow}} & w̃TF \ar[d]^{Z^{w}} \\ A^{osw} \ar@{^{(}->}[r] & A^{sw}}

Proof. Every element of wTF^{o} is also in w̃TF, hence any Z^{w} restricts to a homomorphic expansion Z^{ow} of wTF^{o}. Every element of w̃TF is the result of puncturing – possibly on multiple strands – an element of wTF^{o}, and Z^{w} is required to commute with punctures. Hence any Z^{ow} uniquely extends to a Z^{w}. □

In ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.4] we showed that short arrows – arrows whose head and tail is not separated by any other arrow endings – supported on either strand of V don’t affect whether Z^{w} is a homomorphic expansion. That is, if Z^{w} is a homomorphic expansion and

8Subject to the minor technical condition that the value of the vertex doesn’t contain isolated arrows.

9For convenience we state the Cap Equation phrased for caps at the bottom of strands, hence the difference from the equivalent formulation in \underline{Bar-NatanDancso:WKO2} [WKO2].

DRAFT

a is a linear combination of short arrows, then replacing V by e^{a}V gives rise to another homomorphic expansion. Hence, in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2] we typically assume there are no short arrows in V, this motivates the following definition:

Definition 2.8. A homomorphic expansion Z is v-small if there are no short arrows in the Z-value V of the positive vertex.

As it turns out, the value of the left-punctured vertex is trivial under any v-small homomorphic expansion. This fact will be useful later, so we prove it here.

lem:pV Lemma 2.9. For any v-small homomorphic expansion Z^{w}, Z^{w}( ) = 1, that is, the Z^{w}- value of a left punctured vertex is trivial.

Proof. Recall from ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Proof of Theorem 4.9] that the Z^{w}-value V of the positive (not punctured) vertex can be written as V = e^{b}e^{t}, where b is a linear combination of wheels only and t (denoted uD in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2]) is a linear combination of trees. Puncturing the left strand of V kills all arrow diagrams with tails on the left strand. Diagrams that survive are wheels, and trees all of whose tails are on the right side strand. However, if all tails of a tree are supported on one strand, then the tree is a single arrow, due to TC and the anti-symmetry of the trivalent arrow vertices, thus the only surviving trees are simple arrows directed from right to left. Observe that all of these arrow diagrams commute with each other in A^{sw}(↑_{2}).

Denote the value of the punctured vertex by p_{1}V = e^{p_{1}(b)}e^{p_{1}(t)}. Recall that V must satisfy the Unitarity Equation of Fact ^{\underline{fact:EquationsForZ}}^{\underline{fact:EquationsForZ}} 2.6, so p_{1}V · A_{1}A_{2}(p_{1}V) = 1. Since wheels have only tails, A_{1}A_{2}(p_{1}(b)) = p_{1}(b). Each arrow has one head, so A_{1}A_{2}(p_{1}(t)) = −p_{1}(t). Hence, using commutativity, p_{1}V · A_{1}A_{2}(p_{1}V) = e^{2p_{1}(b)} = 1, which implies that p_{1}(b) = 0. As for p_{1}(t), one can show that there are no arrows pointing from the right to the left strand by a direct computation in degree 1. □

3. Proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1

sec:Proof

subsec:Part1

3.1. Proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1 Part (1). We prove Part 1 in two steps: first verifying the easier “tree level” case, which nonetheless contains the main idea, then in general.

subsec:Part1TreeProof

3.1.1. Tree level proof of Part (1). Let A^{tree} denote the quotient of A^{sw} by all wheels, and let π : A^{sw} → A^{tree} denote the quotient map (cf ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 3.2]). Part (1) of the main theorem is the same as stating that Z^{w} is determined by Z^{u}. Z^{w}, in turn is determined by the values V and C of the positive vertex and the cap ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Sections 4.3 and 4.5], so one only needs to show that V and C are determined by Z^{u}. Proving this “on the tree level” means showing only that π(V) and π(C) are determined by Z^{u}. In particular, observe that since C is a linear combination of products of wheels (Fact ^{\underline{fact:CapIsWheels}}^{\underline{fact:CapIsWheels}} 2.3), we have π(C) = 1, so we only need to show that π(V) is determined by Z^{u}.

B^{u} =

Let B^{u} denote the “buckle” sKTG, as shown on the right (ignore the dotted lines for now). All edges are oriented up, and by the drawing conventions of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.6] all the vertices in the bottom half of the picture are negative and all the ones in the top half are positive. Let B^{w} = a(B^{u}) ∈ w̃TF, and β^{u} := Z^{u}(B^{u}). Note that β^{u} can be represented as a chord diagram on four strands^{10}: use VI relations to move all chord endings to the “middle” of the skeleton,

10The value of β^{u} is computed explicitly to degree four in \underline{Bar-Natan:WKO4} [WKO4].

DRAFT

between the dotted lines on the picture. Hence, we write β^{u} ∈ A^{u}(↑_{4}). Let β^{w} = α(β^{u}), and note that by the compatibility of Z^{u} and Z^{w} we have β^{w} = Z^{w}(B^{w}). We will perform a series of operations on B^{w} and π(β^{w}) to recover π(V) from it.

First, connect (a circuit algebra operation in w̃TF) a positive vertex to the bottom of B^{w}, as shown in Figure ^{\underline{fig:BuckleToV}}^{\underline{fig:BuckleToV}} 12. Then unzip the edge marked by u, and puncture the edges marked e and e′. Then attach a cap (once again a circuit algebra operation) to the thick black end at the bottom. Finally, unzip the capped strand.

u^{u}^{e′}  ^{e}  ^{u}  = K
Figure 12. From the “buckle” β^{w} to the (modified) vertex.  fig:BuckleToV

Call the resulting w-foam K, as shown at the right in Figure ^{\underline{fig:BuckleToV}}^{\underline{fig:BuckleToV}} 12. What is Z^{w}(K)? Due to the homomorphicity of Z, it is obtained from β^{w} by performing the same series of operations in the associated graded: a circuit algebra composition with V, unzip, punctures, circuit algebra composition with C, and disc unzip. Notice that the left strand of that attached vertex got punctured, and hence by Lemma ^{\underline{lem:pV}}^{\underline{lem:pV}} 2.9 the attached value V cancels.^{11} Z^{w}(K) still depends on the value C. At the tree level, since π(C) = 1, π(Z^{w}(K)) can be computed from β^{w} by performing punctures and unzips. Since β^{w} = α(β^{u}), this means that π(Z^{w}(K)) is determined by Z^{u}.

On the other hand, note that the space of chord diagrams on the skeleton of K is the space A(↑_{2}) by the Sorting Lemma (Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.5) and VI. Note also that K is a circuit algebra combination of a vertex, two left-punctured right-capped vertices and an all-red-strings vertex, and the Z^{w}-values of the latter three are trivial. So π(Z^{w}(K)) = π(V) ∈ A^{tree}(↑_{2}). Hence, π(V) is determined by Z^{u} as needed. □

subsec:Part1Proof

3.1.2. Complete proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1 Part (1). In the previous subsection we showed that Z^{u} determines π(V) ∈ A^{tree}(↑_{2}). The proof of Part (1) is completed by the following Lemma:

lem:PiVEnough Lemma 3.1. For any homomorphic expansion Z^{w} of w̃TF set V = Z^{w}( ) and C = Z^{w}(�). Then π(V) ∈ A^{tree}(↑_{2}) determines both V and C uniquely.

Proof. We use a perturbative argument. By contradiction, assume this is not the case, in particular, first assume that there exist V ≠ V′, both of which are vertex values of Z^{u}- compatible homomorphic expansions, such that π(V) = π(V′). Let v denote the lowest degree term of V − V′. Note that v is primitive and v ∈ ker π, so v is a homogeneous linear combination of wheels. By the Unitarity Equation of Fact ^{\underline{fact:EquationsForZ}}^{\underline{fact:EquationsForZ}} 2.6, we have A_{1}A_{2}(v) = −v. Recall that A_{i} reverses the direction of the strand i and multiplies each arrow diagram by

11Any short arrows would also cancel when the right strand is capped.

DRAFT

μ_{23}S_{2}p_{1}(α(Φ))

Figure 13. A concatenated associator.  fig:Concat

(−1) to the number of heads on that strand. Since v has only tails, A_{1}A_{2}(v) = v, so v = −v, so v = 0, a contradiction. Therefore, π(V) determines V uniquely.

Now we show that V determines C uniquely. Assume there are different values C and C′ in A^{sw}(�) so that (V, C) and (V, C′) are both vertex-cap value pairs of Z^{u}-compatible homomorphic expansions. Let c denote the lowest degree term of C − C′, then c is a scalar multiple of a single wheel. The Cap Equation of Fact ^{\underline{fact:EquationsForZ}}^{\underline{fact:EquationsForZ}} 2.6 implies c_{(12)} = c_{1} + c_{2} in A^{sw}(�_{2}).

There is a well-defined linear map ω : A^{sw}(�_{2}) → ℚ[x, y] sending an arrow diagram – which has arrow tails only on each strand – to “x to the power of the number of tails on strand 1, times y to the power of the number of tails on strand 2”. Assume c = αw_{r}, where w_{r} denotes the r-wheel, and α ∈ ℚ. Then 0 = ω(c_{(12)} − c_{1} − c_{2}) = α((x + y)^{r} − x^{r} − y^{r}), so either r = 1 or α = 0. But w_{1} = 0 in A^{sw} by the RI relation, hence α = 0 and thus c = 0, a contradiction. □

subsec:AETFormula

3.2. Proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1 Part (2). In this section we compute V, the value of the vertex, from Φ, the Drinfel’d associator determining Z^{b}, using the construction of Part (1). In Appendix ^{\underline{app:AET}}^{\underline{app:AET}} A we also show that this result translates to the ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET] formula for Kashiwara-Vergne solutions in terms of Drinfel’d associators.

In the computation of V from Φ, as well as later in the paper, we use two facts about Drinfel’d associators. We summarise these in the following Lemma:

lem:PhiFacts Lemma 3.2. Let Φ = Φ(c_{12}, c_{23}) ∈ A^{u}(↑_{3}) be a Drinfel’d associator, where c_{ij} denotes a chord between strands i and j. Let α(Φ) ∈ A^{w}(↑_{3}) denote the image of Φ in arrow diagrams. Then, the following facts hold:

it:Central (1) Φ(x, y) = Φ(x, −x − z), whenever (x + y + z) is central. it:TwoPunctures (2) p_{i}p_{j}α(Φ) = 1, whenever i, j ∈ {1, 2, 3}, i ≠ j, and p_{i} denotes puncture of the i-th

strand.

it:Concat (3) μ_{23} (S_{2}p_{1}(α(Φ))) = 1, where S_{2} stands for orientation switch of strand 2, and μ_{23} is concatenation (multiplication) of strands 2 and 3 – a circuit algebra operation – as

      shown in Figure ^{\underline{fig:Concat}}^{\underline{fig:Concat}} 13.

Proof. Property (^{\underline{it:Central}}^{\underline{it:Central}} 1) follows from the fact that the logarithm of Φ is a Lie series in x and y with no constant term.

To show Property (^{\underline{it:TwoPunctures}}^{\underline{it:TwoPunctures}} 2), recall that if Φ(x, y) is a Drinfel’d associator, then Φ(0, y) = Φ(x, 0) = 1. Therefore p_{1}p_{2}α(Φ) = 1, because p_{1}p_{2}α(c_{12}) = p_{1}p_{2}(a_{12} + a_{21}) = 0. Similar reasoning shows that p_{2}p_{3}α(Φ) = 1. Finally, p_{1}p_{3}α(Φ(c_{12}, c_{23})) = Φ(a_{21}, a_{23}), and since [a_{21}, a_{23}] = 0 by the TC relation, Φ(a_{21}, a_{23}) = 1.

For Property ^{\underline{it:Concat}}^{\underline{it:Concat}} 3, note that p_{1}(α(Φ)) = Φ(a_{21}, −a_{21} − a_{31}). Thus, strands 2 and 3 carry only arrow tails, and these commute by the TC relation, and S_{2}p_{1}(α(Φ)) = Φ(−a_{21}, a_{21} − a_{31}).

DRAFT

1 ^{2}  ^{3} ^{4}  Φ_{132} R_{32} Φ_{(12)34} ^{⟼ 1}^{⟼ e^{a_{23}/2}}^{⟼ 1}^{⟼ Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)})} Φ_{(13)24}^{−1} ^{⟼^{p^{2}α} Φ(a_{23}, a_{43})} Φ_{123}^{−1}^{⟼^{Z^{b}}}^{B^{b} =}

Figure 14. Computing β^{b}. Strands are numbered at the top and multiplication is read from

bottom to top; the rightmost column lists the images of the factors under p_{1}p_{3}α.  fig:BuckleBraid

Furthermore, tails on strand 3 can be pulled to strand 2 through the concatenation, which identifies a_{21} with a_{31}. Therefore, Φ(−a_{21}, a_{21} − a_{31}) = Φ(−a_{21}, 0) = 1. □

To compute V and prove Part (2) of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1, consider once again the w-tangled foam

K on the right of Figure ^{\underline{fig:BuckleToV}}^{\underline{fig:BuckleToV}} 12.

On one hand, Z^{w}(K) can be computed directly from the generators: Z^{w}(K) = C_{1}C_{2}V_{12} ∈ A^{sw}(↑_{2}), since the values of the left-punctured vertices are trivial. Hence, if we know Z^{w}(K), we know V.

On the other hand, we can compute Z^{w}(K), using the compatibility with Z^{u}, as follows. Note that B^{u} is the closure – in the sense of (^{\underline{eq:cl}}^{\underline{eq:cl}} 1) – of the parenthesised braid B^{b} shown in Figure ^{\underline{fig:BuckleBraid}}^{\underline{fig:BuckleBraid}} 14, B^{w} = a(B^{u}). Using the notation β^{u} = Z^{u}(B^{u}), and β^{w} = Z^{w}(B^{w}), and by the compatibility of Z^{w} with Z^{u}, we have

β^{w} = Z^{w}(B^{w}) = α(Z^{u}(B^{u})) = α(β^{u}).

How does Z^{w}(K) differ from β^{w}? To obtain K, a vertex and a cap were attached to B^{w}, two strands were punctured and the cap unzipped, as in Figure ^{\underline{fig:BuckleToV}}^{\underline{fig:BuckleToV}} 12. The Z^{w}-value of the added vertex cancels when its left strand is punctured, however, the value of the cap remains and is unzipped. Thus, in loose notation, Z^{w}(K) = u(C) · p^{2}(β^{w}), where p^{2} denotes the two punctures – we will compute this value explicitly in terms of associators shortly.

To equate the two approaches, we need to express u(C) · p^{2}(β^{w}) as an element of A^{sw}(↑_{2}), by applying the sorting isomorphism φ of Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.5. By doing so, we obtain

C_{1}C_{2}V_{12} = φ(u(C)p^{2}(β^{w})).  (4) eq:BuckleV

Through a careful analysis of the right hand side, this will imply formula (^{\underline{eqn:AET}}^{\underline{eqn:AET}} 3) stated in Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1. In other words, we need to compute

Υ := φ(u(C)p^{2}(β^{w})).

To achieve this, we use that β^{w} = α(β^{u}), and compute β^{u} in terms of the Drinfel’d associator Φ associated to Z^{u}. By the compatibility of Z^{u} and Z^{b}, it is enough to compute β^{b} := Z^{b}(B^{b}). The result can be read from the picture in Figure ^{\underline{fig:BuckleBraid}}^{\underline{fig:BuckleBraid}} 14:

β^{b} = Φ_{(13)24}^{−1}Φ_{132}R_{32}Φ_{123}^{−1}Φ_{(12)34}.

Recall that the cosimplicial notation used in the subscripts show which strands the diagrams are placed on, for example, Φ_{(13)24}^{−1} = Φ^{−1}(c_{12} + c_{32}, c_{24}). Also recall that R = e^{c/2}, so R_{32} = e^{c_{23}/2}.

As β^{u} is the tree closure of β^{b}, it is given by the same formula interpreted as an element of A^{u}(↑_{4}). One then applies α to obtain β^{w} = α(β^{u}). After the vertex and cap attachment,

DRAFT

1  2 1  3 4 2

Figure 15. Strand numbering convention for K and V: arrow endings from strand 1 and 2 of K are “pushed” to strand 1 of V when applying φ, and arrow endings from strands 3 and

4 are pushed to strand 2.  fig:NumberingK

of Figure ^{\underline{fig:BuckleToV}}^{\underline{fig:BuckleToV}} 12, strands 1 and 3 are punctured and strands 2 and 4 are capped, and in this strand numbering, u(C) = C_{24} Therefore, we have

Υ = φ (C_{24} · p_{1}p_{3}α(Φ_{(13)24}^{−1}Φ_{132}R_{32}Φ_{123}^{−1}Φ_{(12)34})) .

Next, we analyse how the puctures and α act on factors of β^{b}. First observe that p_{3}α(R_{32}) = e^{a_{23}/2}, where a_{ij} is a single arrow pointing from strand i to strand j.

Observe that p_{1}p_{3}α(Φ_{123}^{−1}) = p_{1}p_{3}α(Φ_{123}^{−1}) = 1 by Fact (^{\underline{it:TwoPunctures}}^{\underline{it:TwoPunctures}} 2) of Lemma ^{\underline{lem:PhiFacts}}^{\underline{lem:PhiFacts}} 3.2.

Since strands 1 and 3 are both punctured, no arrows can be supported between these two strands, hence p_{1}p_{3}α(Φ_{(12)34}) = Φ(a_{23}, a_{43}).

By Properrty (^{\underline{it:Central}}^{\underline{it:Central}} 1) of Lemma ^{\underline{lem:PhiFacts}}^{\underline{lem:PhiFacts}} 3.2, Φ_{(13)24}^{−1} = Φ^{−1}(c_{(13)2}, c_{24}) = Φ^{−1}(c_{(13)2}, −c_{(13)2} − c_{(13)4}), so p_{1}p_{3}αΦ_{(13)24}^{−1} = Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}). To summarise,

Υ = φ (C_{24} · Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}) · e^{a_{23}/2} · Φ(a_{23}, a_{43})) .

Note that the expression Φ^{−1}(a_{2(13)}, −a_{2(13)} −a_{4(13)})·e^{a_{23}/2} ·Φ(a_{23}, a_{43}) has only arrow tails on strands 2 and 4, and therefore commutes with C_{24} by the T C relation. Hence, by the definition of φ,

Υ = φ (Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}) · e^{a_{23}/2} · Φ(a_{23}, a_{43}) · C_{24})
= φ (Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}) · e^{a_{23}/2} · Φ(a_{23}, a_{43})) · φ(C_{24}).

Furthermore, by the strand numbering convention shown in Figure ^{\underline{fig:NumberingK}}^{\underline{fig:NumberingK}} 15, we have φ(C_{24}) = C_{12}. Therefore,

V_{12} = C_{1}^{−1}C_{2}^{−1}Υ = C_{1}^{−1}C_{2}^{−1}φ (Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}) · e^{a_{23}/2} · Φ(a_{23}, a_{43})) C_{12},

as stated in part (2) of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1. □

In ^{\underline{Bar-Natan:WKO4}} [WKO4] V is computed to degree 4 using the techniques of this section. Matching this result to the Alekseev–Enriquez–Torossian formula of ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET, Theorem 4.] is technical, and not used anywhere else in the paper, hence we defer this to Appendix ^{\underline{app:AET}}^{\underline{app:AET}} A.

subsec:DT

3.3. Proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1 part (3): the double tree construction. Given a homomorphic expansion Z^{u} of sKTG, in Section ^{\underline{subsec:Part1Proof}}^{\underline{subsec:Part1Proof}} 3.1.2 we showed that if there is to exist a homomorphic expansion Z^{w} of w̃TF compatible with Z^{u}, then V = Z^{w}( ) and C = Z^{w}(�), and hence Z^{w} itself, are uniquely determined by Z^{u}. In Section ^{\underline{subsec:AETFormula}}^{\underline{subsec:AETFormula}} 3.2 we proved the formula

DRAFT

(^{\underline{eqn:AET}}^{\underline{eqn:AET}} 3), which in particular gives an explicit expression for the tree level value of V in terms of the Drinfel’d associator associated to Z^{u}. From here on we denote this value V_{β}^{tree} as it is calculated from the Z^{u}-value of the “buckle” graph:

V_{β}^{tree} = φ (Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)})e^{a_{23}/2}Φ(a_{23}, a_{43}))  (5) eq:VtreeAET

It remains to show that there exists an appropriate cap value C = Z^{w}(�) such that C along with the value of the vertex as given in formula (^{\underline{eqn:AET}}^{\underline{eqn:AET}} 3)

Z^{w}( ) =: V_{β} = C_{1}^{−1}C_{2}^{−1}V_{β}^{tree}C_{(12)}  (6) eq:VBeta

determine a homomorphic expansion of w̃TF, which is furthermore compatible with Z^{u}. In particular, in order to show that the a pair (V_{β}, C) defines a homomorphic expansion, one needs to show that the values V_{β} and C satisfy the equations (^{\underline{eq:R4}}^{\underline{eq:R4}} R4), (^{\underline{eq:U}}^{\underline{eq:U}} U) and (^{\underline{eq:C}}^{\underline{eq:C}} C) of Fact ^{\underline{fact:EquationsForZ}}^{\underline{fact:EquationsForZ}} 2.6. We do this in order of difficulty: first the easiest Cap Equation (^{\underline{eq:C}}^{\underline{eq:C}} C), then Unitarity (^{\underline{eq:U}}^{\underline{eq:U}} U) (assuming (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) and compatibility), then Reidemeister 4 (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) (hard). Finally, we prove compatibility, which is easier again, given the machinery developed for (^{\underline{eq:R4}}^{\underline{eq:R4}} R4).

prop:CapEq Proposition 3.3. For any choice of C ∈ A^{sw}(�), the values V_{β} and C verify (^{\underline{eq:C}}^{\underline{eq:C}} C).

Proof. Substituting V_{β} and C into the Cap equation (^{\underline{eq:C}}^{\underline{eq:C}} C), we need to show that

u(C)(u(C))^{−1}(φ(p_{1}p_{3}β^{w}))^{−1}C_{1}C_{2} = C_{1}C_{2},

in A^{sw}( _{2}). We cancel u(C)(u(C))^{−1} on the left, and multiply on the right by C_{1}^{−1}C_{2}^{−1} (this is valid as A( _{2}) is a right A(↑_{2})-module by stacking). Then we only need to show that (φ(p_{1}p_{3}β^{w}))^{−1} = 1 in A(�_{2}). We can multiply on the right by φ(p_{1}p_{3}β^{w}), hence it’s enough to see that 1 = φ(p_{1}p_{3}β^{w}). This, in turn, is clear by the CP relation since all heads are below all tails in the image of φ. □

To address the unitarity equation, we need to set up some notation prove a few basic Lemmas about arrow diagrams.

def:norm Definition 3.4. For an arrow diagram D ∈ A^{sw}(↑_{n}) let D^{∗}, the adjoint of D, be the arrow diagram A_{1}A_{2}...A_{n}(D), where A_{i} denotes the adjoint operation applied to strand i. For a group-like diagram^{12} D = e^{d}, where d ∈ P^{sw}(↑_{n}), denote by ∥D∥ the diagram DD^{∗}, and call it the “norm^{13}” of D.

lem:Norms Lemma 3.5. The adjoint operation and norm satisfy the following basic properties: pr:mult (1) (D_{1}D_{2})^{∗} = D_{2}^{∗}D_{1}^{∗}. pr:exp (2) If d ∈ P^{sw}(↑_{n}) is a primitive arrow diagram, then (e^{d})^{∗} = e^{d^{∗}}.

pr:wheel (3) If d ∈ P^{sw}(↑_{n}) is a wheel, then d^{∗} = d, and ∥e^{d}∥ = e^{2d}. pr:tree (4) If d ∈ P^{sw}(↑_{n}) is a tree, then ∥e^{d}∥ ∈ exp(⟨wheels⟩).

pr:product (5) If d_{1}, d_{2} ∈ P^{sw}(↑_{n}), then ∥e^{d_{1}}e^{d_{2}}∥ = ∥e^{d_{1}}∥∥e^{d_{2}}∥^{e^{d^{∗}_{1}}} = ∥e^{d_{1}}∥∥e^{d_{2}}∥^{e^{−d_{1}}}, where the expo-

nent denotes conjugation.

pr:Ahor (6) If d ∈ P^{hor}(↑_{n}) is a primitive horizontal chord diagram, then ∥e^{α(d)}∥ = 1.

12While the definition of “norm” makes sense for all arrow diagrams, it is most useful, and only used, in 13We use the word “norm” only because it is notationally intuitive; we do not claim that ∥D∥ satisfies

the context of group-like diagrams.

the properties of a norm. The reader might object that it should be called “norm squared”; in our opinion this would clutter up the notation too much.

DRAFT

Proof. Properties (^{\underline{pr:mult}}^{\underline{pr:mult}} 1), (^{\underline{pr:exp}}^{\underline{pr:exp}} 2), and (^{\underline{pr:wheel}}^{\underline{pr:wheel}} 3) are immediate from the definitions. Property (^{\underline{pr:tree}}^{\underline{pr:tree}} 4) follows from the fact that for a tree arrow diagram d, the sum (d + d^{∗}) is a linear combination of wheels by repeated applications of \overrightarrow{ST U} relations.

We verify property (^{\underline{pr:product}}^{\underline{pr:product}} 5) directly:

∥e^{d_{1}}e^{d_{2}}∥ = e^{d_{1}}e^{d_{2}}e^{d_{2}^{∗}}e^{d_{1}^{∗}} = e^{d_{1}}e^{d_{1}^{∗}}e^{−d_{1}^{∗}}e^{d_{2}}e^{d_{2}^{∗}}e^{d_{1}^{∗}} = ∥e^{d_{1}}∥∥e^{d_{2}}∥^{e^{d^{∗}_{1}}}.

To show the second equality, note that by definition e^{d_{1}^{∗}} = e^{−d_{1}}∥e^{d_{1}}∥. By Property (^{\underline{pr:tree}}^{\underline{pr:tree}} 4), ∥e^{d_{2}}∥ ∈ exp(⟨wheels⟩), so ∥e^{d_{2}}∥^{e^{−d_{1}}} ∈ exp(⟨wheels⟩). Since ∥e^{d_{1}}∥ ∈∈ exp(⟨wheels⟩), it acts trivially on ∥e^{d_{2}}∥^{e^{−d_{1}}}.

To prove property (^{\underline{pr:Ahor}}^{\underline{pr:Ahor}} 6) observe that for a single chord t_{ij} ∈ A^{hor}, α(t_{ij}) = a_{ij} + a_{ji}, that is, the sum of arrows from strand i to strand j and vice versa. Therefore, (α(t_{ij}))^{∗} = −α(t_{ij}). Furthermore, if in P^{sw}(↑_{n}), x and y are two primitive arrow diagrams such that x^{∗} = −x and y^{∗} = −y, then by direct computation we also have ([x, y])^{∗} = −[x, y]. Since the set {(t_{ij})}_{1≤i<j≤n} generate P^{hor}(↑_{n}) as a Lie algebra, this implies that (α(d))^{∗} = −α(d) for any d ∈ P^{hor}(↑_{n}). Thus, by direct computation

∥e^{α(d)}∥ = e^{α(d)}e^{(α(d))^{∗}} = e^{α(d)}e^{−α(d)} = 1.
□

The following theorem states that any tree-level homomorphic expansion of w̃TF that is compatible to Z^{u} lifts to a genuine homomorphic expansion of w̃TF.

thm:Unitarity Theorem 3.6. Suppose that Z^{tree} : w̃TF → A^{tree} is a filtered map with Z^{tree}( ) := V^{tree}, such that V^{tree} satisfies the (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) equation, and for any K ∈ sKTG, Z^{tree}(a(K)) = παZ^{u}(K). Then there exists a group-like C ∈ A^{sw}(�) which, along with V := C_{1}^{−1}C_{2}^{−1}V^{tree}C_{12}, defines a homomorphic expansion Z^{w} : w̃TF → A^{sw}.

Proof. Indeed, the values (V, C) uniquely determine Z. To show that Z is a homomorphic expansion, we need to prove that (V, C) satisfy the (^{\underline{eq:R4}}^{\underline{eq:R4}} R4), (^{\underline{eq:C}}^{\underline{eq:C}} C) and (^{\underline{eq:U}}^{\underline{eq:U}} U) equations. The (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) equation only depends on V^{tree}. Since π(V) = π(C_{1}^{−1}C_{2}^{−1}V^{tree}C_{12}) = V^{tree}, therefore V satisfies (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) by assumption. The fact that (V, C) satisfy the Cap (^{\underline{eq:C}}^{\underline{eq:C}} C) equation follows from

Proposition ^{\underline{prop:CapEq}}^{\underline{prop:CapEq}} 3.3.

For the remainder of this proof we denote V^{tree} =: T. The main difficulty is to show that (V, C) satisfy the Unitarity (^{\underline{eq:U}}^{\underline{eq:U}} U) equation, which can be re-stated as ∥V ∥ = 1. Substituting V = C_{1}^{−1}C_{2}^{−1}V^{tree}C_{12}, the (^{\underline{eq:U}}^{\underline{eq:U}} U) equation becomes C_{1}^{−1}C_{2}^{−1}T C_{12}^{2}T^{∗}C_{2}^{−1}C_{1}^{−1} = 1, and by rearranging, this simplifies to

∥T ∥ = (((C_{12})^{T^{−1}})^{−1} C_{1}C_{2})^{2} ,  (7) eq:Unitarity1

where T^{−1} in the exponent denotes conjugation. We need to prove that there exists a value C = e^{c}, which satisfies this equation. Note that ∥T ∥ is an exponential of wheels in A^{sw}(↑_{2}), so this makes semantic sense. By Fact ^{\underline{fact:semidirect}}^{\underline{fact:semidirect}} 2.2, ∥T_{12}∥ is identified with an element of exp(cyc⟨x, y⟩) and thus represented by an exponential in cyclic words in two variables. Let f(x, y) ∈ cyc⟨x, y⟩ denote the logarithm of ∥T ∥. In turn, c ∈ cyc⟨x⟩ ≅ ℚ[[x]], thus we write c = c(x).

We claim that

(C_{12})^{T^{−1}} = (exp(c(x + y)))^{T^{−1}} = T exp(c(x + y))T^{−1} = exp(c(log(e^{x}e^{y}))).

DRAFT

The last equality follows from the fact that T satisfies the (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) equation; this is explained in detail in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2], Section 4.4, in particular in and around Fig. 19.

With this reduction in place (^{\underline{eq:Unitarity1}}^{\underline{eq:Unitarity1}} 7) reduces to

e^{f(x,y)} = (e^{c(log(e^{x}e^{y}))}e^{c(x)}e^{c(y)})^{2} ,

and since all exponents (wheels) commute, this is equivalent to

f(x, y) = 2(c(x) + c(y) + c(log(e^{x}e^{y}))).  (8) eq:Unitarity2

which we need to solve for c ∈ ℚ[[x]].

To do so, we use that Z^{tree}(a(K)) = παZ^{u}(K), specifically for the associator graph K = ϕ shown on the right. Recall that Z^{u}(ϕ) is a Drinfel’d associator Φ. On the other hand, α(ϕ) is a circuit algebra product of four vertices, therefore

α(Φ) = Z^{tree}(a(ϕ)) = T_{12,3}^{−1}T_{12}^{−1}T_{23}T_{1,23}.

Rearranging, we obtain

T_{12}T_{12,3}α(Φ) = T_{23}T_{1,23}.

Applying the “norm” to both sides, keeping in mind that ∥α(ϕ)∥ = 1 by Lemma ^{\underline{lem:Norms}}^{\underline{lem:Norms}} 3.5 property (^{\underline{pr:Ahor}}^{\underline{pr:Ahor}} 6), and using the exponent notation for the adjoint action on a norm, we get

∥T_{12}∥∥T_{12,3}∥^{T_{12}^{−1}} = ∥T_{23}∥∥T_{1,23}∥^{T_{23}^{−1}}.  (9) eq:UnitarityMidProof

Recall that ∥T_{12,3}∥ = exp(f(x + y, z)), ∥T_{23}∥ = exp(f(y, z)), ∥T_{1,23}∥ = exp(f(x, y + z)). As before, we have

∥T_{12,3}∥^{T_{12}^{−1}} = (exp(f(x + y, z)))^{T_{12}^{−1}} = T_{12} exp(f(x + y, z))T_{12}^{−1} = exp(f(log(e^{x}e^{y}), z)),

since T satisfies the (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) equation. Similarly,

∥T_{1,23}∥^{T_{23}^{−1}} = (exp(f(x, y + z)))^{T_{23}^{−1}} = exp(f(x, log(e^{y}e^{z}))).

  Thus, Equation (^{\underline{eq:UnitarityMidProof}}^{\underline{eq:UnitarityMidProof}} 9) reduces to

e^{f(x,y)}e^{f(log(e^{x}e^{y}),z)} = e^{e^{f(y,z)}}e^{f(x,log(e^{y}e^{z}))},

and since all the exponentials (wheels) commute, this is equivalent to

f(x, y) = f(y, z) + f (x, log(e^{y}e^{z})) − f (log(e^{x}e^{y}), z)  (10)

The fact that there exists a solution c(x) to Equation ^{\underline{eq:Unitarity2}}^{\underline{eq:Unitarity2}} 8 follows from this equation via a

short cohomology calculation as in ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET, Proposition 27], and ^{\underline{AlekseevTorossian:KashiwaraVergne}} [AT, Appendix A, Proof of

Theorem 2.8]. □

In order to use Theorem ^{\underline{thm:Unitarity}}^{\underline{thm:Unitarity}} 3.6, it is necessary to show that the value V^{tree} constructed in Section ^{\underline{subsec:Part1TreeProof}}^{\underline{subsec:Part1TreeProof}} 3.1.1 satisfies the (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) equation; unfortunately this is hard. The equation (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) is an equality between different circuit algebra – in fact, planar algebra – compositions of crossings. Hence, the proof would be much easier if Z^{u} were to be a circuit algebra – or planar algebra – map. This unfortunately makes no sense, as sKTG is not a circuit or planar algebra but a different, more complicated algebraic structure.

The reader might ask, why work with a space as inconvenient as sKTG instead of, say, a planar algebra of trivalent tangles? The answer is that the existence of a homomorphic expansion is a highly non-trivial property, and in particular ordinary trivalent tangles do not have one. Even without trivalent vertices, ordinary tangles, or u-tangles, do not have have

DRAFT

a homomorphic expansion as a planar algebra^{14}. Parenthesized tangles (a.k.a. q-tangles) \underline{LeMurakami:Universal, Bar-Natan:NAT} [LM, BN2] do have homomorphic expansions, yet in fact these are almost equivalent to

sKTG ^{\underline{Thurston:KTG, Bar-NatanDancso:KTG, Dancso:KIforKTG}} [T, BND1, Da].

Nonetheless, we can harness the power of planar algebras in a less direct way to prove the R4 equation, by building a new definition for Z^{w}. Since we will be working with two constructions (eventually proving that they lead to the same expansion), it will be necessary to distinguish them, at least temporarily.

• From here on we denote the value V^{tree} arising from the “buckle” construction of

      Section ^{\underline{subsec:Part1TreeProof}}^{\underline{subsec:Part1TreeProof}} 3.1.1 by V_{β}^{tree}. Assuming the conditions of Theorem ^{\underline{thm:Unitarity}}^{\underline{thm:Unitarity}} 3.6 hold, there is a

unique homomorphic expansion Z_{β}^{w} for which π(Z^{w}( )) = V_{β}^{tree}.

• In a new construction, we map classical trivalent tangles into sKTG via a “double tree” map, and use this to define a partial expansion Z_{ξ}^{w} for the a-images of classical trivalent tangles. This in particular defines a new vertex value Z_{ξ}^{w}( ) = V_{ξ}.

• After some technical lemmas, we show that Z_{ξ}^{w} satisfies the compatibilty Z^{w}(a(K)) =

      αZ^{u}(K) for all K ∈ sKTG: see Proposition ^{\underline{prop:uwCompatibility}}^{\underline{prop:uwCompatibility}} 3.14.

• We show that restriction of Z_{ξ}^{w} to a(uT T) is a planar algebra map (see Theorem ^{\underline{thm:PAMap}}^{\underline{thm:PAMap}} 3.17), and thus Z_{ξ}^{w} satisfies the (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) equation.

• We show that the two constructions yield the same vertex value on the tree level:

      π(V_{ξ}) = V_{β}^{tree}. This is done in Lemma ^{\underline{lem:TwoConstructions}}^{\underline{lem:TwoConstructions}} 3.19. Thus, by Lemma ^{\underline{lem:PiVEnough}}^{\underline{lem:PiVEnough}} 3.1, if the two

constructions produce homomorphic expansions, those expansions are the same.

• Putting this together, V_{β}^{tree} satisfies (^{\underline{eq:R4}}^{\underline{eq:R4}} R4) and the compatibility condition of Theorem ^{\underline{thm:Unitarity}}^{\underline{thm:Unitarity}} 3.6, and thus exists a unique homomorphic expansion Z^{w} = Z_{ξ}^{w} = Z_{β}^{w}, compatible with Z^{u}. We prove this in detail in Theorem ^{\underline{thm:ZwExpansion}}^{\underline{thm:ZwExpansion}} 3.22 below.

subsubsec:ZwDef

3.3.1. The new construction: Z_{ξ}^{w}. We start by defining (classical, or usual) trivalent tangles, denoted uT T:

uT T := PA ⟨ , , , | R1^{s}, R2, R3, R4 | S_{e}, u_{e} ⟩

Here PA stands for planar algebra: an algebra over the operad of planar tangles, that is, an algebraic structure similar to a circuit algebra, except with planar wiring diagrams. (See \underline{DHR:CircAlg} [DHR1, Section 3.1] for a detailed definition. This is a slightly more simple-minded notion than the original use of the term in ^{\underline{Jones:PlanarAlgebras}} [J], in particular we do not use checkerboard shadings.)

+e  u_{e}^{−}

The elements of uT T are usual – that is, classical – trivalent tangles with ordered ends (the ordering is assumed to be counterclockwise from bottom left, unless otherwise stated), and signed vertices with a total ordering of edges at each vertex. There are no “virtual crossings” in planar algebras. The relations are the usual Reidemeister relations which make sense in this context (R1^{s}, R2, R3 and R4). The planar algebra uT T is equipped with auxiliary orientation switch and edge unzip operations. Edge unzips are defined for edges

14We only mention that the planar algebra of u-tangles does not have a homomorphic expansion Z^{t} so as to explain why we are not using one. This said, the non-existance of Z^{t} is easy to prove: by an explicit calculation in degree 2 one shows that there is no linear combination of chord diagrams that can serve as Z^{t}(�), satisfying the R3 relation.

DRAFT

↦^{�} T  T^{−}^{−}^{−}  ^{−}^{−}^{−}^{+}^{+}  ^{+} ^{+}

Figure 16. The double tree map: connect the ends of T by two binary trees (hence “double tree”), as shown. Note that the tree on the left always crosses over the tree on the right,

and all edges of both trees are oriented towards T.  fig:dt

=  T T  Ť T^{1}^{2}^{3}^{4} T T  ↦^{κ}  ↦^{u}  ↦^{p}

Figure 17. The cap attachment, unzips and punctures. While these operations are applied in A^{sw} – there are arrows on these skeleta – for simplicity the figure only shows the effect on

the skeleton.  fig:pCu

that connect a positive and a negative vertex in as shown in the figure on the right. The planar algebra uT T does not have a homomorphic expansion.

We define a double tree map � : uT T → sKTG, as in Figure ^{\underline{fig:dt}}^{\underline{fig:dt}} 16. The map � depends on two choices of binary trees: in Figure ^{\underline{fig:dt}}^{\underline{fig:dt}} 16 we chose a particular example. It is important that, regardless of the choice of trees, the “left side” tree crosses over the “right side” tree. We will demonstrate that in fact the choice of trees becomes irrelevant after some post-compositions,

see Lemma ^{\underline{lem:TreeChange}}^{\underline{lem:TreeChange}} 3.7.

Working towards a construction of Z_{ξ}^{w}, we post-compose � with the following sequence of maps, which are explained in the paragraph below:

T ∈ uT T ⟶^{�} sKTG ⟶^{Z^{u}} A^{u}(�(T)) ⟶^{α} A^{sw}(�(T)) ⟶^{κ,u,p} A^{sw}(Ť) ≅^{φ} A^{sw}(T).  (11) eq:dt

Here T stands for an arbitrary tangle in uT T. The double tree map � maps T into sKTG, and by applying Z^{u} one obtains a value in A^{u}, namely a chord diagram on the skeleton of �(T). We denote the space of chord diagrams on this skeleton by A^{u}(�(T)). Now α maps this to arrow diagrams on the skeleton of �(T), that is, to A^{sw}(�(T)). In order to revert the skeleton back to that of T, we apply some operations in A^{sw}: a cap attachment κ, unzips and punctures (as shown in Figure ^{\underline{fig:pCu}}^{\underline{fig:pCu}} 17 and explained below), resulting in a slightly modified version of the desired skeleton, denoted Ť. Finally, we use that A^{sw}(Ť) ≅ A^{sw}(T) via the sorting isomorphism φ of Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.5, and hence we obtain a value in A^{sw}(T), as needed, which, we will later see, is almost Z^{w}(a(T)). (Although the punctured strands connect in a single binary tree, VI relations can be used as part of the sorting isomorphism.)

DRAFT

The cap attachment, unzip and puncture operations are done in the order shown in Figure ^{\underline{fig:pCu}}^{\underline{fig:pCu}} 17. First attach a cap – a capped strand with no arrows on it – to the end of the right vertical strand in α(�(T)): this is a circuit algebra operation in A^{sw}. If T has n ends, perform (n − 1) consecutive disc unzips on the capped strand, as shown in Figure ^{\underline{fig:pCu}}^{\underline{fig:pCu}} 17. Then puncture the left hand tree, for example by puncturing the left vertical strands marked “1, 2,...” in Figure ^{\underline{fig:pCu}}^{\underline{fig:pCu}} 17 (these punctures also affect the connecting diagonal strands, as in Figure ^{\underline{fig:punctures}}^{\underline{fig:punctures}} 5). Note that since the punctured tree had originally crossed over the capped tree, these crossings become virtual after puncturing, hence the last equality in Figure ^{\underline{fig:pCu}}^{\underline{fig:pCu}} 17.

Denote the composition of the maps and operations shown in Equation (^{\underline{eq:dt}}^{\underline{eq:dt}} 11) by ξ, that is,

ξ := φ ∘ p ∘ u ∘ κ ∘ α ∘ Z^{u} ∘ � : uT T → A^{sw}.  (12) eq:xidef

Then, ξ(T) ∈ A^{sw}(T). We first show that ξ(T) is well-defined, that is, it doesn’t depend on the choice of binary trees in �(T).

lem:TreeChange Lemma 3.7. The choice of binary trees in the double tree construction does not affect ξ(T).

Proof. Any binary tree can be changed into any other binary tree via a sequence of “I to H” moves, as shown on the right. Hence, it is enough to analyze how an I to H move on one of the trees affects the value of Z^{u}(�(T)), and prove that the difference vanishes after the caping, unzip, and puncture operations.

2 ^{1}  u^{2}

Suppose τ_{1} and τ_{2} are two binary trees which differ by a single I to H move, and let �_{τ_{1}} and �_{τ_{2}} denote the two resulting doubletree maps, assuming the “other side tree” is unchanged. The I to H move can be realised by inserting^{15} an associator, followed by unzipping the edge marked ‘1’ on the right, then the edge marked ‘2’. By the homomorphicity of Z^{u}, the values Z^{u}(�_{τ_{2}}(T)) and Z^{u}(�_{τ_{1}}(T)), only differ in an inserted horizontal chord associator Φ on the three strands involved, we indicate this by writing Z^{u}(�_{τ_{2}}(T)) = Z^{u}(�_{τ_{1}}(T)) ∗ Φ. If the I to H move was done on the left side tree, then all the strands involved are later punctured, killing any arrow diagram that lived on them by the TF relation. As a result, the only surviving part of Φ is its constant term, 1, and the resulting values of ξ are equal.

If the I to H move is done on the right side tree, then the all participating strands are capped and disk unzipped. If α(Φ) is immediately adjacent to the caps, then it cancels by the CP relation. However, it is a priori possible that there are other arrow ending separating Φ from the caps. Note that in A^{u}, any chord endings can be can be commuted from below the associator to above, using V I relations and the invariance property of chord diagrams shown in Figure ^{\underline{fig:Invariance}}^{\underline{fig:Invariance}} 18 ^{\underline{Bar-Natan:NAT}} [BN2, Lemma 3.4]. Thus, one can assume that α(Φ) is adjacent to the caps and hence cancels. This concludes the proof. □

There is an action of ℤ/nℤ on elements of uT T with n ends, by cyclic permutations of the ends. The following lemma will be useful later in proving that Z^{w} is a planar algebra map; we present it now because its proof is similar to that of Lemma ^{\underline{lem:TreeChange}}^{\underline{lem:TreeChange}} 3.7.

lem:CyclicPerm Lemma 3.8. The map ξ is invariant under cyclic permutation of the ends of T.

15See \underline{Bar-NatanDancso:WKO2} [WKO2, Section 4.6] for a detailed description of the tangle insertion operation in sKTG.

DRAFT

=^{D}  ^{D}  ^{D}  = ^{D}  =^{D ∈ im α}  ^{D ∈ im α}^{+}  ^{+}  ^{+}  ^{+}  ^{+}  ^{+}

Figure 18. The invariance property of chord diagrams on the left, the head invariance and tail invariance properties of arrow diagrams in the middle and right. Here D denotes a chord or arrow diagram on any skeleton; for tail invariance there is a restriction that D ∈ im α (with α : A^{u} → A^{sw}). The box labelled “+” denotes a sum of incoming chords or arrows, whose other ends are in the same place. The equalities are understood locally: there may be

other skeleton components and chords/arrows elsewhere, which coincide on both sides.  fig:Invariance

T  T  T =  ∗ ∗ �(σT) �(T)
Figure 19. Double tree construction for cyclically permuted ends of T.  fig:welldef

Proof. To show that ξ(T) is invariat under cyclic permutations of ends of T, it is enough to show that ξ(T) does not change when the rightmost end of T is moved to the far left (denote this by σT), as shown in Figure ^{\underline{fig:welldef}}^{\underline{fig:welldef}} 19.

The rightmost picture of Figure ^{\underline{fig:welldef}}^{\underline{fig:welldef}} 19 is equivalent as sKTGs to �(σT). It differs from �(T) in three ways:

• the binary trees connecting the ends of T are different;

• two tree branches (marked with ∗ in Figure ^{\underline{fig:welldef}}^{\underline{fig:welldef}} 19) are connected to the trunk from the opposite side: that is, these trivalent vertices have opposite cyclic orientation;

• one tree branch has a kink in it. As before, we need to analyse how Z^{u}(�(σT)) differs from Z^{u}(�(T)), and show that the

difference vanishes after the puncture, cap and unzip operations.

To achieve this, we transform �(σT) into �(T) using tangle insertions. First, cancel the kink by inserting an opposite kink I_{1} on the same strand, as shown in Figure ^{\underline{fig:welldef2}}^{\underline{fig:welldef2}} 20 in blue^{16}. As Z^{u} is compatible with insertion, the Z^{u} values will differ by the value of a kink: a chord diagram on the one strand involved. Later in the process this strand is punctured, so the value of the kink cancels by the TF relation.

Similarly, switching the side that the tree branches are attached on amounts to inserting twists I_{2} and I_{3}, and unzipping the connecting edges, also shown in Figure ^{\underline{fig:welldef2}}^{\underline{fig:welldef2}} 20. Each of these

16Or grey in black and white print.

DRAFT

unzips  T  �(T) T^{unzip} �(σT) with inserts: I_{2}  I_{3} I_{1}
Figure 20. The difference between �(T) and �(σT), understood via insertions.  fig:welldef2
[figure: T; ∈; ↦; �(T); ∈; ↦; �(T); ∈; ↦; κ�(T); ∈; ↦; T̂; ∈; uT T; �; ξ; sKTG; a; Z^{u}; (1); w̃TF; κ; Z^{w}; (2); w̃TF; p∘u; Z^{w}; (3); w̃TF; Z^{w}; A^{u}(�(T)); α; A^{sw}(�(T)); κ∘Ĉ; A^{sw}(κ�(T)); p∘u; A^{sw}(T̂); φ; (4); A^{sw}(T)]

Figure 21. Comparing ξ and Z_{ξ}^{w}, assuming that Z_{ξ}^{w} exists.  fig:BigCompat

operations changes the value of Z^{u} by inserting the value of a twist, which is e^{c/2} for any Z^{u}, where c denotes a single chord between the appropriate strands ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Lemma 4.14]. Applying α maps this to e^{(a_{L}+a_{R})/2}, where a_{L} and a_{R} denote horizontal left and right arrows, respectively. On the left side tree, this cancels after punctures, as before. On the right side tree, the strand directly underneath the twist is capped and uzipped, and hence the value of the twist cancels by the CP relation.

Now observe that the right side picture of Figure ^{\underline{fig:welldef2}}^{\underline{fig:welldef2}} 20 only differs from �(T) in the choices of binary trees, which do not change the value of ξ by Lemma ^{\underline{lem:TreeChange}}^{\underline{lem:TreeChange}} 3.7. □

The following lemma clarifies the relationship between the map ξ and the (partial) homomorphic expansion Z_{ξ}^{w} that we’re aiming to construct:

lem:Compatibility Lemma 3.9. If there exists a homomorphic expansion Z_{ξ}^{w} for w̃TF compatible with Z^{u}, and T ∈ uT T is a tangle with n ends, then Z_{ξ}^{w}(a(T)) = ξ(T) · (C^{−1})^{n}, where C = Z( ), and ξ(T) is multiplied by C^{−1} at each tangle end of T, as in Figure ^{\underline{fig:ZTCheck}}^{\underline{fig:ZTCheck}} 22.

Proof. Assume there exists a homomorphic expansion Z_{ξ}^{w} compatible with Z^{u}. We use, as in Figure ^{\underline{fig:BigCompat}}^{\underline{fig:BigCompat}} 21, the homomorphicity of Z_{ξ}^{w} and its compatibility with Z^{u} to show that ξ(T) = Z_{ξ}^{w}(Ť), where Ť is as in Equation ^{\underline{eq:dt}}^{\underline{eq:dt}} 11 and shown in Figure ^{\underline{fig:ZTCheck}}^{\underline{fig:ZTCheck}} 22 on the left.

DRAFT

Z^{w}(T)^{Ť}^{T}  ^{Z^{w}(T)} ^{↦^{φ}}^{↦^{Z^{w}}}  ^{Z^{w}(T)} = ^{ξ(T)}^{C} ^{C}^{C}  ^{C} ^{C} ^{C}  ^{C^{−1}}^{C^{−1}}  ^{C^{−1}}
Figure 22. Computing Z_{ξ}^{w}(Ť) and Z_{ξ}^{w}(T).  fig:ZTCheck

If the diagram in Figure ^{\underline{fig:BigCompat}}^{\underline{fig:BigCompat}} 21 commutes, then for any T ∈ uT T and any Z^{u}-compatible Z_{ξ}^{w}, we have φ(Z_{ξ}^{w}(Ť)) = ξ(T). Since Z_{ξ}^{w} is a circuit algebra homomorphism, Z_{ξ}^{w}(Ť) can be obtained from Z_{ξ}^{w}(T) by attaching the Z_{ξ}^{w}-value of a left-punctured right-capped vertex at

each tangle end, as illustrated in Figure ^{\underline{fig:ZTCheck}}^{\underline{fig:ZTCheck}} 22. By Lemma ^{\underline{lem:pV}}^{\underline{lem:pV}} 2.9 we have Z_{ξ}^{w}( ) = 1, so the only additions are C values at each capped end, as shown in Figure ^{\underline{fig:ZTCheck}}^{\underline{fig:ZTCheck}} 22. This can then be interpreted as a value in A^{sw}(T) via the sorting isomorphism φ of Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.5. This implies the statement of the Lemma.

It remains to show that the diagram in Figure ^{\underline{fig:BigCompat}}^{\underline{fig:BigCompat}} 21 commutes. The square (1) is the assumed the compatibility of Z^{u} and Z_{ξ}^{w}. In square (2), recall the map κ denotes the circuit algebra operation of attaching a cap at the bottom right end of the w-foam. The map Ĉ denotes the circuit algebra operation which attaches a value C = Z( ) at the end of the strand. Thus, the commutativity of square (2) is implied by the homomorphicity of Z_{ξ}^{w} with respect to circuit algebra composition (as a binary operation). The square (3) is commutative due to the homomorphicity of Z_{ξ}^{w} with respect to punctures and disc unzips.

T  = In imα T C  C

The commutativity of the heptagon (4) would be true by definition, if not for the map Ĉ (multiplication by the cap value). We show that, in fact, the value C cancels after punctures, by a property of arrow diagrams in the image of α, called tail-invariance, shown in Figure ^{\underline{fig:Invariance}}^{\underline{fig:Invariance}} 18 (see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2], Remark 3.14 and early in Section 3.3). In the current situation tail invariance means that the value C, which has only arrow tails, can be moved from one tangle end to the other, as shown on the right. Consequently, C cancels when the left strand is punctured. □

rem:CapOrientations Remark 3.10. In Lemma ^{\underline{lem:Compatibility}}^{\underline{lem:Compatibility}} 3.9 we assume by convention that all tangle ends of T are oriented upwards (towards T). If k tangle ends are oriented down, the corresponding cap values appear with their orientations switched: Z_{ξ}^{w}(aT) = ξ(T) · (C^{−1})^{n−k}(S(C)^{−1})^{k}.

cor:ZTreefromxi Corollary 3.11. If there exists a homomorphic expansion Z^{w} for w̃TF compatible with Z^{u}, then π(V) = π(ξ(�)), where V is the Z^{w}-value of the vertex, and π is the tree projection.

Next, we aim to show that Z_{ξ}^{w} is compatible with Z^{u}. This requires a technical lemma, in which we compute the ξ-value of a vertical strand:

lem:xiofstrand Lemma 3.12. For a single un-knotted strand, ξ(↷) = α(ν^{1/2}), where ν ∈ A^{u}(↷) denotes the Kontsevich integral of the un-knot^{17}.

17The value of ν was conjectured in \underline{Bar-NatanGaroufalidisRozanskyThurston:WheelsWheeling} [BGRT] and proven in \underline{Bar-NatanLeThurston:TwoApplications} [BLT]. Note that ν involves wheels only.

DRAFT

+^{−} ^{+}^{−} ↦^{�}  ^{=}  ↦^{Z^{u}}  ^{Φ^{−1}}^{b}  ^{n}^{Φ}^{Φ^{−1}}^{S_{3}(Φ^{−1})}^{R}^{Φ}

Figure 23. The double tree map composed with Z^{u}, applied to a single strand. To compute Z^{u}(�(↷)), we write �(↷) as a composition of generators; this requires first expressing it

as a bottom-top tangle. See \underline{Bar-NatanDancso:WKO2} [WKO2, Proposition 4.13] for details.  fig:dtstrand

Proof. We apply � to ↷, as shown in Figure ^{\underline{fig:dtstrand}}^{\underline{fig:dtstrand}} 23, and comupte Z^{u}(�(↷)) using the finite generation property of sKTG and the homomorphicity of Z^{u}. In ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 5.2] we gave an algorithm for writing any sKTG as an sKTG-composition of generators (the primary operation in sKTG is tangle insertion, see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Figure 22]). Feeding �(↷) into this algorithm, one needs to “curve up” one strand as in Figure ^{\underline{fig:dtstrand}}^{\underline{fig:dtstrand}} 23, in this case the strand on the right (the choice of strand doesn’t affect the outcome).

The chord diagram Z^{u}(�(↷)) is shown in Figure ^{\underline{fig:dtstrand}}^{\underline{fig:dtstrand}} 23, expressed in terms of the generators of sKTG described in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Proposition 4.13]: the value Φ of the associator graph, which is a Drinfel’d associator; the value R = e^{c/2} of the twist graph, where c is a single chord; and the values n and b of the noose and balloon graphs, respectively.

In ξ(↷), Z^{u} is followed by α, a cap attachment, unzips and punctures. As explained in \underline{Bar-NatanDancso:WKO2} [WKO2, Section 4.6], we have α(b) = e^{a/2}α(ν)^{1/2}, and α(n) = e^{−a/2}α(ν)^{1/2}. Both factors are local arrow diagrams on a single strand and hence commute with all other arrow endings. Hence the exponential part of n cancels by the CP relation, and the wheels part α(ν)^{1/2} can be moved to the bottom left end by the tail invariance property (as in the last paragraph of the proof of Lemma ^{\underline{lem:Compatibility}}^{\underline{lem:Compatibility}} 3.9), where it cancels given punctures. Subsequently, the associator

α(S_{3}(Φ^{−1})) cancels by Fact (^{\underline{it:Concat}}^{\underline{it:Concat}} 3) of Lemma ^{\underline{lem:PhiFacts}}^{\underline{lem:PhiFacts}} 3.2.

Taking these cancellations into account, the value puκαZ^{u}�(↷) ∈ A^{sw} is shown in Figure \underline{fig:dtstrand2}^{\underline{fig:dtstrand2}} 24 and explained below. Recall that α maps a chord to the sum of its two possible orientations. However, when one supporting strand is punctured, only one of these orientations survive. Hence, for example, p_{2}(α(R_{23})) = (e^{a_{32}/2}). Figure ^{\underline{fig:dtstrand2}}^{\underline{fig:dtstrand2}} 24 shows a schematic picture of puκαZ^{u}�(↷) with exponentials and associators indicated by single arrows. Recall that Φ ∈ A^{hor}(↑_{3}) can be written as a power series in any two of the three generators of A^{hor}(↑_{3}): c_{12}, c_{23} and c_{13}. For example, Φ(c_{12}, c_{23}) = Φ(c_{12}, −c_{12} − c_{23}). For each associator we chose the presentation in which puκα(Φ) is of the simplest form.

(1) The top associator of Figure ^{\underline{fig:dtstrand}}^{\underline{fig:dtstrand}} 23, after applying a VI relation, is written as Φ_{13(24)}^{−1} in the strand numbering of Figure ^{\underline{fig:dtstrand2}}^{\underline{fig:dtstrand2}} 24. We write this in terms of c_{13} and c_{1(24)} = c_{12}+c_{14}. Observe that p_{1}α(c_{13}) = a_{31} and p_{1}p_{2}α(c_{1(24)}) = a_{41}, thus

p_{1}p_{2}αΦ^{−1}(c_{13}, −c_{13} − c_{1(24)}) = Φ^{−1}(a_{31}, −a_{31} − a_{41}).

DRAFT

5^{1} ^{2}  ^{3} ^{4}^{α(Φ)_{324}}  ^{(1)}^{(2)}^{(3)}^{(4)}^{(5)}^{e^{a/2}}^{(6)}  ^{α(Φ)_{12(34)}}^{α(Φ^{−1})_{234}}^{(e^{a_{32}/2}) from α(R)}^{α(Φ^{−1})_{134}}^{+}^{α(ν^{1/2})}

Figure 24. The value of puκαZ^{u}�(↷): the numbering (1) through (6) refer to the para-

graphs where each component is explained.  fig:dtstrand2

This is reflected in Figure ^{\underline{fig:dtstrand2}}^{\underline{fig:dtstrand2}} 24 in drawing only the a_{31} and a_{41} arrows. Now note that by the VI relation at the top right vertex, a_{41} = −a_{51}. The local arrow diagrams e^{a/2} and α(ν^{1/2}) on strand 5 commute with arrow tails, and therefore a_{51} can be pulled to the left, where by another VI relation, a_{41} = −a_{51} = −a_{31}. Thus, Φ^{−1}(a_{31}, −a_{31} − a_{41}) = Φ^{−1}(a_{31}, 0) = 1. Note the similarity to Fact (^{\underline{it:Concat}}^{\underline{it:Concat}} 3) of Lemma ^{\underline{lem:PhiFacts}}^{\underline{lem:PhiFacts}} 3.2.

(2) Second from top, by the same argument, we have

p_{1}p_{2}α(Φ_{324}) = p_{1}p_{2}α(Φ(c_{23}, c_{24})) = Φ(a_{32}, a_{42}) = Φ(a_{32}, −a_{32}) = 1.

(3) For the exponential,

p_{1}p_{2}α(R_{23}) = p_{1}p_{2}α(e^{c_{23/2}}) = e^{a_{32}/2}.

(4) Using the same reasoning again,

p_{1}p_{2}α(Φ_{234}^{−1}) = p_{1}p_{2}α(Φ^{−1}(c_{23}, −c_{23} − c_{24})) = Φ^{−1}(a_{32}, −a_{32} − a_{42}) = Φ^{−1}(a_{32}, 0) = 1.

Note that here we use that the arrow tail of a_{32} commutes with the arrow tails of the exponential.

(5) By Fact (^{\underline{it:TwoPunctures}}^{\underline{it:TwoPunctures}} 2) of Lemma ^{\underline{lem:PhiFacts}}^{\underline{lem:PhiFacts}} 3.2,

p_{1}p_{2}α(Φ)_{12(34)} = 1.

(6) Finally, move the top exponential e^{a/2} to strands 3 and 2, using the VI relation at both vertices. At the top left vertex, the arrow tails move freely from strand 5 to strand 3. The heads commute with α(ν), and at the top right vertex they are killed on strand 4 due to the CP relation, and on strand 2 they acquire a negative sign due to opposite orientations. Hence, (e^{a_{55}/2}) = (e^{−a_{32}/2}), and this cancels α(R).

In summary, ξ(↷) = α(ν^{1/2}), as claimed. □

As an aside, Lemma ^{\underline{lem:xiofstrand}}^{\underline{lem:xiofstrand}} 3.12 enables a quick computation of the even part of C = e^{c} = Z_{ξ}^{w}(�). Recall that c is a linear combination of wheels: c = ∑_{n=2}^{∞} γ_{n}w_{n}. Let c = c_{0} + c_{1}, where c_{0}

DRAFT

K  =  K
Figure 25. For K ∈ sKTG, �(K) is K inserted into �(↷).  fig:InsertK

denotes the even part of c (sum of all even wheels), and c_{1} denotes the odd part, that is, c = c_{0} + c_{1}. Let C_{0} = e^{c_{0}}, the even part of C. Corollary ^{\underline{cor:CapValue}}^{\underline{cor:CapValue}} 3.13 below shows in particular that the even part of the C is independent of the choice of Z^{b} (that is, the choice of Drinfel’d associator) and Z^{u}.

cor:CapValue Corollary 3.13. If C = Z_{ξ}^{w}(�), and C_{0} is the even part of C, then C_{0} = α(ν^{1/4}), regardless of the choice of expansion Z^{u} used to construct Z_{ξ}^{w}.

Proof. By Lemma ^{\underline{lem:Compatibility}}^{\underline{lem:Compatibility}} 3.9 and Remark ^{\underline{rem:CapOrientations}}^{\underline{rem:CapOrientations}} 3.10, we have Z_{ξ}^{w}(↷) = C^{−1}ξ(↷)S(C^{−1}). Note that

S(w_{2k}) = w_{2k} and S(w_{2k+1}) = −w_{2k+1}, and hence S(C) = e^{c_{0}−c_{1}}. Also, by homomorphicity, Z_{ξ}^{w}(↷) = 1. Thus, by Lemma ^{\underline{lem:xiofstrand}}^{\underline{lem:xiofstrand}} 3.12, 1 = e^{c_{0}+c_{1}}α(ν^{1/2})e^{c_{0}−c_{1}}, and therefore α(ν^{1/2}) = e^{2c_{0}}, which gives C_{0} = e^{c_{0}} = α(ν^{1/4}).

Next we prove that Z_{ξ}^{w} is indeed compatible with Z^{u}:

prop:uwCompatibility Proposition 3.14. For any K ∈ sKTG, Z_{ξ}^{w}(a(K)) = α(Z^{u}(K)).

Proof. Note that sKTG ⊆ uT T, and for K ∈ sKTG, �(K) can be obtained by inserting K into the top strand of �(↷): see Figure^{\underline{fig:InsertK}}^{\underline{fig:InsertK}} 25. Since Z^{u} is compatible with insertions, Z^{u}(�(K)) can be obtained by Z^{u}(K) inserted into Z^{u}(�(↷)). Through the sequence of α, capping, puncturing, φ and multiplications by C^{−1}, all of Z^{u}(�(↷)) cancels, as in Lemma ^{\underline{lem:xiofstrand}}^{\underline{lem:xiofstrand}} 3.12. Note that the cancellations still go through despite the fact that α(Z^{u}(K)) is inserted on the top strand: this follows from the fact that α(Z^{u}(K)) is in the α-image of A^{u}, and the appropriate “commutativity” property holds in A^{u}. Hence, Z_{ξ}^{w}(K) = Z_{ξ}^{w}(K) as required. □

Now, we show that Z_{ξ}^{w} it is a planar algebra homomorphism on a(uT T). This is technically challenging, but it implies the R4 equation immediately. For the proof, it will be necessary to know the behaviour of Z^{u} with respect to edge deletions. When an edge e of a knotted trivalent graph K ∈ sKTG is deleted, the two vertices at each end of e cease to be vertices. The associated graded operation on chord diagrams deletes skeleton edge e, and chord diagrams with any chord endings on e are set equal to 0.

fact:ZuDelete Fact 3.15. Z^{u} commutes with edge deletions up to a possible correction term of e^{±c/4}ν^{1/2} depending on the position of the edge, as in Figure ^{\underline{fig:ZuDelete}}^{\underline{fig:ZuDelete}} 26.

Proof sketch. In ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.6.1] Z^{u} is constructed from an invariant Z^{old} by adding vertex normalizations, as shown in Figure ^{\underline{fig:ZuDelete}}^{\underline{fig:ZuDelete}} 26. Note that the top two diagrams in the figure differ from ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Figure 29] in a single edge orientation switch, which switches the vertex

DRAFT

Z^{u} d  d  d  d Z^{u} d  ·e^{\frac{c}{4}} ν^{\frac{1}{2}} d  d Z^{u}  ·e^{− \frac{c}{4}} ν^{\frac{1}{2}} d Z^{u}^{−} Z^{u}^{+} Z^{u}  Z^{u}^{−} Z^{u}^{+}^{e^{\frac{c}{4}}}^{ν^{− \frac{1}{4}}}^{ν^{− \frac{1}{4}}}^{ν^{\frac{1}{4}}}^{ν^{\frac{1}{4}}}  ^{ν^{− \frac{1}{4}}}^{ν^{− \frac{1}{4}}} ^{ν^{− \frac{1}{4}}}^{ν^{− \frac{1}{4}}}^{ν^{− \frac{1}{4}}}  ^{e^{− \frac{c}{4}} ν^{− \frac{1}{2}}}  ^{e^{\frac{c}{4}} ν^{− \frac{1}{2}}}^{e^{− \frac{c}{4}}}^{ν^{\frac{1}{4}}}^{e^{\frac{c}{4}}}^{ν^{\frac{1}{4}}}^{e^{− \frac{c}{4}}}^{ν^{− \frac{1}{4}}}

Figure 26. In \underline{Bar-NatanDancso:WKO2} [WKO2, Section 4.6.1] Z^{u} is constructed from an invariant Z^{old} by applying vertex normalizations, which depend on vertex signs: these are shown along the top horizontal arrow of each diagram (see also \underline{Bar-NatanDancso:WKO2} [WKO2, Figure 29]). It follows Z^{u} is only homomorphic up to a correction term when deleting the top edge of a positive vertex (first in the total ordering around the vertex) or the bottom edge of a negative vertex: see the top two diagrams. In other edge deletions the normalizations cancel, and hence Z^{u} is homomorphic with respect

to these edge deletions, as for example in the bottom two diagrams.  fig:ZuDelete

sign and accordingly the normalization^{18}. In fact, Z^{old} commutes with edge deletions ^{\underline{Dancso:KIforKTG}} [Da, Proposition 6.7], so the edge deletion error (and hence, the correction term) for Z^{u} arises from the vertex normalisations implemented, as shown in Figure ^{\underline{fig:ZuDelete}}^{\underline{fig:ZuDelete}} 26. □

rmk:wDelete Remark 3.16. There is also an “edge delete” operation of w̃TF: this is not required for the finite presentation of w̃TF or A^{sw}, but it is necessary for the proof of Theorem ^{\underline{thm:PAMap}}^{\underline{thm:PAMap}} 3.17 below. When deleting an edge in w̃TF – which can be either a tube or a string – the vertices at either end^{19} cease being vertices. The associated graded operation d_{e} : A^{sw} → A^{sw} deletes the skeleton edge e and sends any arrow diagram with arrow endings on the deleted strand to zero. The crucial fact we need is that edge delete operations for chord and arrow diagrams

18The point of the normalization is to make Z^{u} commute with unzips. The reader might wonder, why normalize so that the expansion respects unzips, rather than deletions? The answer is that for finite generation of knotted trivalent graphs, unzips are crucial but deletions are not.

19It is also possible to delete a capped edge.

DRAFT

⊔^{3}  T_{1}^{1} ^{2} ^{3} ^{4}  T_{2}^{5} ^{6} ^{7}^{,}  T_{2}^{4}  ^{1} ^{2} T_{1}^{1} ^{2} ^{3}  ^{c_{3}}  ^{1} ^{2}  ^{3} T^{1} ^{2} ^{3} ^{4} ^{5} T
Figure 27. Basic planar algebra operations: disjoint union and contraction.  fig:AtomicPAOps

are compatible via the map α, which is immediate from the definitions:

\xymatrix{A^{u} \ar[r]^{α} \ar[d]^{d_{e}} & A^{sw} \ar[d]^{d_{e}} \\ A^{u} \ar[r]^{α} & A^{sw}}

thm:PAMap Theorem 3.17. The map Z_{ξ}^{w} is a planar algebra map on a(uT T).

Proof. Planar algebra operations can be written as compositions of two simpler, basic operations: disjoint unions and contractions. In the disjoint union of two tangles T_{1} and T_{2}, the ends of T_{1} ⊔ T_{2} are ordered by declaring that the ordered ends of T_{1} come first, followed by the ordered ends of T_{2}. The contraction operation c_{i} applies to any tangle with at least i + 1 ends: it acts by joining the i-th and (i + 1)-st ends of T and re-numbering the rest, resulting in a tangle with two ends less. Both operations are shown in Figure ^{\underline{fig:AtomicPAOps}}^{\underline{fig:AtomicPAOps}} 27.

Thus, we only need to show that Z_{ξ}^{w} commutes with these two operations, that is, Z_{ξ}^{w}(T_{1} ⊔ T_{2}) = Z_{ξ}^{w}(T_{1}) ⊔ Z_{ξ}^{w}(T_{2}), and Z_{ξ}^{w}(c_{i}(T)) = c_{i}(Z_{ξ}^{w}(T)). Note that the right sides of these equalities make sense: arrow diagrams on the skeleta of a(uT T), where Z_{ξ}^{w} takes values, also form a planar algebra, and in particular disjoint union and concatenation of arrow diagrams is well defined.

Disjoint unions. We need to compute ξ(T_{1} ⊔ T_{2}), where T_{1}, T_{2} ∈ uT T. The value �(T_{1} ⊔ T_{2}) is shown in Figure ^{\underline{fig:DisjUni}}^{\underline{fig:DisjUni}} 28. The binary trees in � can be chosen arbitrarily by Lemma ^{\underline{lem:TreeChange}}^{\underline{lem:TreeChange}} 3.7: Figure ^{\underline{fig:DisjUni}}^{\underline{fig:DisjUni}} 28 shows the most convenient trees for this proof.

Observe that �(T_{1} ⊔ T_{2}) can be obtained as an sKTG by inserting �(T_{1}) and �(T_{2}) into a simpler sKTG, denoted H, as shown in the same figure (up to orientation switches which don’t impact what follows and will be ignored). Hence, Z^{u}(�(T_{1} ⊔ T_{2})) is given by inserting Z^{u}(�(T_{1})) and Z^{u}(�(T_{2})) into Z^{u}(H).

One could compute Z^{u}(H) explicitly using the same algorithm as before, but we can avoid this work, as follows. All chords in Z^{u}(H) can be assumed to be located in the rectangle shown in Figure ^{\underline{fig:DisjUni}}^{\underline{fig:DisjUni}} 28 (using VI relations, if necessary). During the computation of ξ both supporting strands are punctured, and therefore p^{2}α(Z^{u}(H)) = 1. This implies that ξ(T_{1} ⊔T_{2}) = ξ(T_{1})⊔ξ(T_{2}), and it follows via Lemma ^{\underline{lem:Compatibility}}^{\underline{lem:Compatibility}} 3.9 that Z_{ξ}^{w}(T_{1} ⊔T_{2}) = Z_{ξ}^{w}(T_{1})⊔Z_{ξ}^{w}(T_{2}).

Contractions. Proving that Z_{ξ}^{w} commutes with contractions is more involved. By Lemma ^{\underline{lem:CyclicPerm}}^{\underline{lem:CyclicPerm}} 3.8, we can assume that the ends contracted are the last (rightmost) two ends of the n ends of T. Hence we will drop the subscript from c_{i} and denote this operation simply by c.

DRAFT

H T_{1}  T_{2}  ^{�(T_{1})} ^{�(T_{2})}^{insert} ^{insert}  ^{Z^{u}(H)}^{all chords}

Figure 28. The double tree map applied to a disjoint union of uT T -s is the same as inserting the double tree of each individual uT T into the sKTG H. In Z^{u}(H) all chords can be pushed

into the rectangle shown, using VI relations when necessary.  fig:DisjUni

[figure: uT T; �; c; (1); sKTG; Z^{u}; d^{3}; (2); A^{u}; α; (·e^{− \frac{c}{4}} ν^{\frac{1}{2}} )∘d^{3}; (3); A^{sw}; κ; (·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} ))∘d^{3}; (4); A^{sw}; u^{n−1}; (·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} ))∘d^{3}; (5); A^{sw}; p^{n−2}; (·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} ))∘d^{3}; (6); uT T; �; sKTG; Z^{u}; A^{u}; α; A^{sw}; κ; A^{sw}; u^{n−3}; A^{sw}; p^{n−2}]

diagram continues... [figure: A^{sw}; p^{2}; (·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} ))∘d^{3}; (7); A^{sw}; φ; A^{sw}; ·C^{−(n−1)}SC^{−1}; (·α(ν^{− \frac{1}{2}} ))∘c; (8); A^{sw}; c; A^{sw}; φ; A^{sw}; ·C^{−(n−2}); A^{sw}]

Figure 29. Summary of the proof that Z_{ξ}^{w} commutes with contractions: Z_{ξ}^{w} is the compo-

sition along the entire top and entire bottom horizontal edge of the diagram.  fig:ContractCommutes

We need to show that Z_{ξ}^{w}(cT) = cZ_{ξ}^{w}(T), for any T ∈ uT T. Since Z_{ξ}^{w} is given by the composition of many maps, this can be restated as the commutativity of the perimeter of a large diagram (shown in Figure ^{\underline{fig:ContractCommutes}}^{\underline{fig:ContractCommutes}} 29), which in turn can be broken down to its smaller parts. Throughout this proof, let T ∈ uT T denote an arbitrary trivalent tangle.

c  ^{d^{3}}  ^{d^{3}}^{(2)}  ^{·e^{− \frac{c}{4}} ν^{\frac{1}{2}}}^{(1)} T T ⟶^{�}  ^{⟶^{Z^{u}}}  T ⟶^{�} T  T^{⟶^{Z^{u}}} T

Square (1). This square (shown schematically on the right) plays out in uT T and sKTG, and commutes by inspection, as shown on the right. The three strands to be deleted are indicated by broken lines. Therefore, d^{3}�(T) = �c(T).

Square (2). Square (2) is also on the right: for the Z^{u}- values skeleta are indicated but chords are not shown. To prove that square (2) commutes, we use the properties of Z^{u} with respect to deleting edges in sKTG, as

stated in Fact ^{\underline{fact:ZuDelete}}^{\underline{fact:ZuDelete}} 3.15 and Figure ^{\underline{fig:ZuDelete}}^{\underline{fig:ZuDelete}} 26.

Only one of the three edge deletions requires a correction term: this is the edge marked with ∗ in the diagram on the left. This edge ends in a

DRAFT

·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )^{·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )} T T^{d^{3}}  ⟶^{·C} ⟶^{·C}  ⟶^{u^{n−3}} ⟶^{u^{n−1}}^{(4)}  ^{(5)}  ^{(6)} ⟶^{p^{n−2}} ⟶^{p^{n−2}}^{·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )}  ^{∗} T T^{d^{3}}  T T^{d^{3}}  ^{·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )}^{d^{3}} T T

Figure 30. The squares (4) (5) and (6). Strands to be deleted are drawn in dashed lines

throughout. The ∗ denotes a cap of interest: see the proof paragraph on square (5).  fig:456

e^{−c/4}ν^{1/2} inserted at the place of the vertex, where c stands for a single chord. In square (2), this correction term appears at the bottom right corner of the square, where the two ends of T are contracted (see in the diagram showing skeleta in Figure ^{\underline{fig:ContractCommutes}}^{\underline{fig:ContractCommutes}} 29). In summary: (d^{3}Z^{u}�(T)) · (e^{− \frac{c}{4}} ν^{\frac{1}{2}} ) = Z^{u}�c(T).

T  T ⟶^{α} T  T ⟶^{α}^{d^{3}}  ^{d^{3}}^{·e^{− \frac{c}{4}} ν^{\frac{1}{2}}}  ^{·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )}^{(3)}

Square (3). Square (3) is essentially the commutativity of edge deletions stated in Remark ^{\underline{rmk:wDelete}}^{\underline{rmk:wDelete}} 3.16, combined with applying α to the correction term. So we have (d^{3}αZ^{u}�(T)) · (e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )) = αZ^{u}�c(T).

Square (4). Squares (4), (5), and (6) are shown in detail in Figure ^{\underline{fig:456}}^{\underline{fig:456}} 30. Square (4) plays out in A^{sw} and it is commutative as the deletions and the cap attachments (denoted by κ) affect different strands: see the diagram on the right. Therefore, (d^{3}καZ^{u}�(T)) · (e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )) = καZ^{u}�c(T).

Square (5). The only difference between d^{3}∘u^{n−1} and u^{n−3}∘d^{3} is what happens to arrows on the caped skeleton edge marked by ∗ in Figure ^{\underline{fig:456}}^{\underline{fig:456}} 30. Following the diagram right and down, this edge is unzipped n − 1 times, then the last two of its daughter edges are deleted. On the other hand, following the diagram down and right, the same edge is unzipped n − 3 times. The results of these compositions are the same by definition of the unzip and delete operations. Thus, we have

(d^{3}u^{n−1}καZ^{u}�(T)) · (e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )) = u^{n−3}καZ^{u}�c(T).

Square (6).The deletions and punctures occur on different strands, as shown in Figure ^{\underline{fig:456}}^{\underline{fig:456}} 30, hence these operations commute commute. One detail to note is that when a tube strand is deleted at a “tube-and-string” vertex, all that is left is a string (as in the case of puncturing the tube at a tube-string vertex, see Figure ^{\underline{fig:punctures}}^{\underline{fig:punctures}} 5). In summary:

(d^{3}p^{n−2}u^{n−1}καZ^{u}�(T)) · (e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )) = p^{n−2}u^{n−3}καZ^{u}�c(T).

DRAFT

T S  A  unzips T

Figure 31. Computing the top left corner of Square 7, Step 1: �(T) can be expressed as the sKTG denoted S inserted into the sKTG denoted A, followed by unzips, as shown. Z^{u}

respects insertions, hence computing Z^{u}(A) determines the value of Z^{u}(�(T)) outside of S.  fig:Square7Comm

unzips  Z^{u} A  A  D b  n β^{u}^{β^{u}} S_{3}(Φ^{−1})

Figure 32. Computing the top left corner of Square 7, Step 2: computing Z^{u}(A). The sKTG A can be obtained by inserting the buckle sKTG twice into a simpler sKTG, and unzipping, as shown on the left. The value of the buckle was computed in Figure \underline{fig:BuckleBraid} \underline{fig:BuckleBraid} 14. Using this value—denoted β^{u}—and the algorithm in \underline{Bar-NatanDancso:WKO2} [WKO2, Section 5.2], one computes Z^{u}(A).

The result is denoted D and shown on the right.  fig:Square7Comm2

d^{3} T T  T  ⟶^{≅} ⟶^{p^{2}}^{(7)}  T^{c}^{·α(ν^{− \frac{1}{2}} )}^{·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )}  ^{≅}  T

Pentagon (7). The pentagon (7) is shown on the left. This is the most delicate part of the proof. We first show that – for the specific input of p^{n−2}u^{n−1}καZ^{u}�(T) – the pentagon (7) commutes up to a single possible multiplicative error on the contracted (u-shaped) strand, and later prove that this error is necessarily 1.

To begin, a better understanding of the arrow diagram p^{n−2}u^{n−1}καZ^{u}�(T) in the top left corner is necessary. All of the operations performed on

T, with the exception of Z^{u}, are “easy” in the sense that we have a complete understanding of their effect. Z^{u} is “hard”, but we can compute the relevant part of its value using the finite generation of sKTG (^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Proposition 4.13]). The computation is shown in Figures

\underline{fig:Square7Comm}^{\underline{fig:Square7Comm}} 31 and ^{\underline{fig:Square7Comm2}}^{\underline{fig:Square7Comm2}} 32 and their captions.

In summary, Z^{u}(�(T)) is given by inserting Z^{u}(A) into the chord diagram D of Figure ^{\underline{fig:Square7Comm2}}^{\underline{fig:Square7Comm2}} 32. Now we need to analyze what happens when one applies α, the cap attachment, unzips and punctures to this value: this is an exercise similar to what has been done for Lemma ^{\underline{lem:xiofstrand}}^{\underline{lem:xiofstrand}} 3.12.

DRAFT

U U  ^{T}  ^{e^{−a/2}}^{T}  ^{(7)}  ^{T} ⟶^{p^{2}}  U  ^{α(Φ^{−1})}^{e^{a/2}}^{α(Φ)}^{S}  ^{≅ φ}^{c}^{T}  ^{e^{−a/2}}^{φ(U)}  ^{T}^{≅ φ}  ^{φ(U)}^{·e^{− \frac{a}{2}} α(ν^{\frac{1}{2}} )}^{d^{3}}  ^{·α(ν^{−1/2})}^{β^{w}}^{pu(β^{w})}  ^{pu(β^{w})}  ^{+}^{α(ν^{1/2})}  ^{p(β^{w})}^{α(ν^{1/2})}^{S_{2}αu(b)}  ^{S_{2}αu(b)}^{φ(p(β^{w}))}^{φ(pu(β^{w}))}^{S_{2}αu(b)}
Figure 33. The more detailed picture of the pentagon 7.  fig:Square7Big

The result is shown in Figure ^{\underline{fig:Square7Big}}^{\underline{fig:Square7Big}} 33, and explained below. First note that the n value in D of Figure ^{\underline{fig:Square7Comm2}}^{\underline{fig:Square7Comm2}} 32 cancels after punctures by the tail-invariance property (Figure ^{\underline{fig:Invariance}}^{\underline{fig:Invariance}} 18), as in the last paragraph of the proof of Lemma ^{\underline{lem:Compatibility}}^{\underline{lem:Compatibility}} 3.9; so does the bottom Φ^{−1} in D , by Fact (^{\underline{it:Concat}}^{\underline{it:Concat}} 3) of

Lemma ^{\underline{lem:PhiFacts}}^{\underline{lem:PhiFacts}} 3.2. These components are not shown in Figure ^{\underline{fig:Square7Big}}^{\underline{fig:Square7Big}} 33.

Working downwards from the top left of the pentagon in Figure ^{\underline{fig:Square7Big}}^{\underline{fig:Square7Big}} 33, the three edge deletions cancel both buckle (β^{w}) values. The value b value at the top of the diagram D is pulled down across the vertex using a V I relation: this has the same effect as an unzip and an orientation switch on the second stand (as this is oriented downwards). The resulting value Sαu(b) cancels by the following reasoning – essentially by definition of unzips and orientations switches – which is illustrated in Figure ^{\underline{fig:KillAnArrow}}^{\underline{fig:KillAnArrow}} 34:

Given an arrow ‘a’ ending on strand ‘e’, unzipping e produces a sum of two arrows a_{1} +a_{2}: one ending on each daughter strand. Reversing the orientation of the first daughter strand gives −a_{1} + a_{2}. Contracting the two daughter strands to form a U-shape identifies a_{1} and a_{2}, making a_{1} − a_{2} vanish.

DRAFT

+  ^{−} = 0
Figure 34. Unzip, switch orientation and connect kills arrows on the strand.  fig:KillAnArrow

T  ⟼^{φ}  T ^{+} ^{−}  ⟼^{c}  T ^{+} ^{−}  = 0
Figure 35. The cancellation of φ(pu(β^{w})).  fig:puBuckle

This is exactly what happens to the arrow diagram S_{2}αu(b) (just under the tangle T in Figure ^{\underline{fig:Square7Big}}^{\underline{fig:Square7Big}} 33) after the edge deletions or contraction, hence this component cancels it in both directions of the pentagon (7).

The bottom arrow on the left is the Sorting Isomorphism φ of Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.5.

On the other hand, working from the top left corner of the pentagon (7) to the right: from

Section ^{\underline{subsec:AETFormula}}^{\underline{subsec:AETFormula}} 3.2, using the strand numbering convention of Figure ^{\underline{fig:NumberingK}}^{\underline{fig:NumberingK}} 15, we have

p_{1}p_{3}β^{w} = Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}) · e^{a_{23}/2} · Φ(a_{23}, a_{43}).

This is shown in the enlarged rectangle at the top right corner of Figure ^{\underline{fig:Square7Big}}^{\underline{fig:Square7Big}} 33. The same is true for pu(β^{w}) at the bottom, except with more unzips.

From here, the first downward arrow applies the Sorting Isomorphism φ of Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.5, followed by the contraction and correction term along the second downward arrow. At the bottom of the diagram, φ(pu(β^{w})) cancels altogether after contraction, in a similar fashion to Figure ^{\underline{fig:KillAnArrow}}^{\underline{fig:KillAnArrow}} 34. Namely, φ and the contraction annihilates any arrow ending on the diagonal red strand, or the double capped strand on the right, as shown in Figure ^{\underline{fig:puBuckle}}^{\underline{fig:puBuckle}} 35 for the diagonal red strand. This cancels each factor of pu(β^{w}).

Of the top φ(p(β^{w})) shown in the enlarged rectangle, the Φ^{−1} component cancels by the same contraction argument; only the exponential and the Φ component remains. The arrow in the exponent of e^{a/2} switches sign due to the reversed orientation: this is the e^{−a/2} component at the bottom of the pentagon in Figure ^{\underline{fig:Square7Big}}^{\underline{fig:Square7Big}} 33. After contraction, the Φ component gives rise to a “local” arrow diagram on a single strand shown in Figure ^{\underline{fig:mu}}^{\underline{fig:mu}} 36 and denoted μ.

In summary, we see that the pentagon (7) commutes if and only if μ = α(ν), and otherwise commutes up to a localised error on the contracted strand, of value α(ν)^{−1}μ.

DRAFT

Φ(a_{12}, a_{32})  =:^{3}^{2}^{1}  ^{μ}
Figure 36. The Φ component of φ(p(β^{w})) after contraction.  fig:mu

T  T ⟶^{·C^{−(n−1)}SC^{−1}} T  T ⟶^{·C^{−(n−2)}}^{c}^{·α(ν^{− \frac{1}{2}} )}  ^{c}^{(8)}

Square (8). Finally, for square (8) we need to check that C^{−1}S(C^{−1}) = α(ν^{−1/2}). Note that the orientation switch negates odd wheels and preserves even wheels, therefore in C^{−1}S(C^{−1}) the odd part of C^{−1} cancels, and C^{−1}S(C^{−1}) = C_{0}^{−2} = α(ν^{−1/2}) by Corollary ^{\underline{cor:CapValue}}^{\underline{cor:CapValue}} 3.13. This verifies Square (8).

We have therefore shown that Z^{w} commutes with contraction up to an error α(ν)^{−1}μ on the contracted strand. It remains to show that this error is 1. This follows from the facts that Z^{w}(↑) = 1, and that Z^{w} commutes with disjoint unions:

1 = c(Z^{w}(↑↓)) = α(ν)^{−1}μZ^{w}(c(↑↓)) = α(ν)^{−1}μ · Z^{w}(↷) = α(ν)^{−1}μ.

This completes the proof. □

Note that as a side result we have proven the following curious fact about associators:

Proposition 3.18. For any Φ horizontal chord associator, μ defined from Φ as in Figure ^{\underline{fig:mu}}^{\underline{fig:mu}} 36, and ν the Kontsevich integral of the unknot, μ = α(ν) in A^{sw}. □

The value V = Z_{ξ}^{w}( ) differs, on first glance, from the value V_{β} constructed in Section ^{\underline{subsec:Part1}}^{\underline{subsec:Part1}} 3.1. In the next lemma we show that in fact V = V_{β}: this serves both as a reality check, and as a technical tool for showing – in Theorem ^{\underline{thm:ZwExpansion}}^{\underline{thm:ZwExpansion}} 3.22 – that Z_{ξ}^{w} satisfies the Cap equation.

lem:TwoConstructions Lemma 3.19. The two vertex values from the buckle and the double tree constructions coincide: V_{β} = V.

Proof. Notice that �(�) – as in Figure ^{\underline{fig:DTandBuckle}}^{\underline{fig:DTandBuckle}} 37 – can be obtained from simpler sKTGs by inserting the buckle B^{u} into �(↷) followed by an unzip. We computed ξ(↷) in Lemma ^{\underline{lem:xiofstrand}}^{\underline{lem:xiofstrand}} 3.12. Since Z^{u} is compatible with insertions and unzips, Z^{u}(�) can be computed by inserting the buckle value β^{u} (computed in Section ^{\underline{subsec:AETFormula}}^{\underline{subsec:AETFormula}} 3.2) into Z^{u}(�(�)).

To compute ξ(�) we then apply α and the cap, unzip and puncture operations. Because β^{w} = α(β^{u}) is local (is confined to the skeleton of the inserted B^{u} graph) and is in the image of α, all other arrow endongs commute with it by the head- and tail-invariance properties of A^{sw} (Figure ^{\underline{fig:Invariance}}^{\underline{fig:Invariance}} 18). Therefore, all of the cancellations in the computation of ξ(↷) still occur, and ξ(�) is as shown in Figure ^{\underline{fig:DTandBuckle}}^{\underline{fig:DTandBuckle}} 37. Writing this in A^{sw}(↑_{2}) we obtain φ(p_{1}p_{3}(β^{w}))u(α(ν^{1/2})).

To obtain V = Z_{ξ}^{w}( ), one multiplies ξ(�) at each end by C^{−1} or S(C^{−1}) depending on orientation, as shown in Figure ^{\underline{fig:DTandBuckle}}^{\underline{fig:DTandBuckle}} 37 on the right. Thus,

V = C_{1}^{−1}C_{2}^{−1}φ(p_{1}p_{3}β^{w})u(αν^{1/2})u(S(C^{−1}).

DRAFT

e  puκαZ^{u} u_{e} B^{u} in �(↷)  �(�)  α(ν^{1/2}) p_{1}p_{3}(β^{w})  ^{φ(p_{1}p_{3}(β^{w}))}^{u(α(ν^{1/2}))}^{C^{−1}}  ^{C^{−1}}  ^{S(C^{−1})}

Figure 37. On the left we show how to obtain �(�) via an unzip from B^{u} insterted into

�(↷). From this we compute ξ(�), and finally V on the right.  fig:DTandBuckle

On the other hand, from Section ^{\underline{subsec:AETFormula}}^{\underline{subsec:AETFormula}} 3.2, we have:

V^{β} = C_{1}^{−1}C_{2}^{−1}φ(p_{1}p_{3}β^{w})u(C).

Thus, we need to show that:

C_{1}^{−1}C_{2}^{−1}φ(p_{1}p_{3}β^{w})u(αν^{1/2})u(S(C^{−1}) = C_{1}^{−1}C_{2}^{−1}φ(p_{1}p_{3}β^{w})u(C),

in A^{sw}(↑_{2}). Multiplying with (φ(p_{1}p_{3}β^{w})))^{−1} C_{1}C_{2} on the left (bottom) and by u(S(C)) on the right (top), this simplifies:

u(αν^{1/2}) = u(C)u(S(C)).

Since unzips commute with orientation switches, it is sufficient to prove that

CS(C) = α(ν^{1/2}).

Recall that in CS(C) all odd wheels cancel, hence CS(C) = (C_{0})^{2}, where C_{0} denotes the even part of C. Indeed, by Corollary ^{\underline{cor:CapValue}}^{\underline{cor:CapValue}} 3.13, C_{0}^{2} = α(ν^{1/2}). □

cor:SameResult Corollary 3.20. The “buckle” amd “double tree” constructions lead to the same result, that is, Z_{ξ}^{w} = Z_{β}^{w}.

Proof. Since any homomorphic expansion Z^{w} of w̃TF is uniquely determined by Z^{w}( ),

this is immediate from Lemma ^{\underline{lem:TwoConstructions}}^{\underline{lem:TwoConstructions}} 3.19.  □

The next lemma implies that V satisfies the Unitarity (^{\underline{eq:U}}^{\underline{eq:U}} U) equation:

lem:Unzip Lemma 3.21. The map Z_{ξ}^{w} commutes with strand unzips in uT T.

Proof. We first note that in uT T unzip is only defined for internal edges, that is, edges which end in a vertex^{20} at both ends. By construction, Z_{ξ}^{w} is a composition of several maps. We show that edge unzips commute with every one of these, hence with Z_{ξ}^{w}.

The � map involves only the tangle ends, which are unchanged by unzipping an internal strand, hence these operations commute. The homomorphic expansion Z^{u} commutes with edge unzips, as shown in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.6]. The map α commutes with edge unzips by definition. Cap attachments, cap unzips and the isomorphism φ commute with internal

20In fact, there are further restrictions, eg the two vertices must be of opposite signs, but this is not important for the proof.

DRAFT

unzips for the same reason as �, because they are performed at the tangle ends while unzip is internal. This completes the proof. □

Finally, the following Theorem completes the proof of part Part (3) of the Main Theo-

rem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1:

thm:ZwExpansion Theorem 3.22. The map Z_{ξ}^{w} : w̃TF → A^{w} is the unique homomorphic expansion of w̃TF, which is compatible with Z^{u} in the sense of the commutative diagram (^{\underline{eq:MainDiag}}^{\underline{eq:MainDiag}} 2).

Proof. By Proposition ^{\underline{lem:Compatibility}}^{\underline{lem:Compatibility}} 3.9, Z_{ξ}^{w} is compatible with Z^{u}. By Part (1) (Section ^{\underline{subsec:Part1}}^{\underline{subsec:Part1}} 3.1) and Corollary ^{\underline{cor:SameResult}}^{\underline{cor:SameResult}} 3.20, Z_{ξ}^{w} = Z_{β}^{w} and uniquely determined by Z^{u}.

  To show that Z_{ξ}^{w} is a homomorphic expansion, by Fact ^{\underline{fact:EquationsForZ}}^{\underline{fact:EquationsForZ}} 2.6 and Theorem ^{\underline{thm:ExtendRestrict}}^{\underline{thm:ExtendRestrict}} 2.7, one only

needs to verify that it satisfies the (^{\underline{eq:R4}}^{\underline{eq:R4}} R4), Unitarity (^{\underline{eq:U}}^{\underline{eq:U}} U) and Cap (^{\underline{eq:C}}^{\underline{eq:C}} C) equations of Fact ^{\underline{fact:EquationsForZ}}^{\underline{fact:EquationsForZ}} 2.6. Of these, R4 follows from the fact that Z_{ξ}^{w} is a planar algebra map, Theorem ^{\underline{thm:PAMap}}^{\underline{thm:PAMap}} 3.17. The Unitarity equation (^{\underline{eq:U}}^{\underline{eq:U}} U) is equivalent to the statement that Z_{ξ}^{w} commutes with strand unzips \underline{Bar-NatanDancso:WKO2} [WKO2, Section 4.3], and hence it is satisfied by Lemma ^{\underline{lem:Unzip}}^{\underline{lem:Unzip}} 3.21.

This leaves the Cap Equation, □

Appendix A. The Alekseev–Enriquez–Torossian formula

app:AET

This appendix is mainly for readers familiar with the Alekseev–Enriquez–Torossian formula for Kashiwara–Vergne solutions in terms of Drinfel’d associators ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET].

For a quick re-cap of ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET] notions, let 𝔩𝔦𝔢_{2} denote the degree completed free Lie algebra on two generators x and y. Let 𝔱𝔡𝔢𝔯_{2} denote tangential derivations of this Lie algebra, that is, derivations d with the property that d(x) = [x, a_{1}] and d(y) = [y, a_{2}], where a_{1}, a_{2} ∈ 𝔩𝔦𝔢_{2}. Let TAut_{2} := exp(𝔱𝔡𝔢𝔯_{2}) denote the group of tangential automorphisms of 𝔩𝔦𝔢_{2}.

There is a linear “interpretation map” (not a map of Lie algebras) θ : 𝔩𝔦𝔢_{2}^{2} → 𝔱𝔡𝔢𝔯_{2}, sending a pair (a_{1}, a_{2}) to the derivation d given by d(x) = [x, a_{1}], d(y) = [y, a_{2}]. The kernel of this map consists only of pairs of the form (αx, βy) for α, β constants. A one-sided inverse η : 𝔱𝔡𝔢𝔯_{2} → 𝔩𝔦𝔢_{2}^{2} which sends a tangential derivation to a pair whose first component has no linear x term and second component has no y term. We denote the exponential of θ by Θ : U(𝔩𝔦𝔢_{2}^{2})_{exp} → TAut_{2}. For an element (e^{λ_{1}}, e^{λ_{2}}) ∈ U(𝔩𝔦𝔢_{2}^{2})_{exp}, we have G = Θ((e^{λ_{1}}, e^{λ_{2}})) ∈ TAut_{2} given by G(x) = e^{−λ_{1}}xe^{λ_{1}}, and G(y) = e^{−λ_{2}}ye^{λ_{2}}. Just as θ is not a Lie algebra map, Θ is not a group homomorphism: composition in TAut_{2} is not given by piecewise multiplication of the conjugators. However, θ and Θ present a convenient way to denote tangential derivations and tangential automorphisms as pairs in 𝔩𝔦𝔢_{2}^{2} and (exp(𝔩𝔦𝔢_{2}))^{2}, respectively.

Given a Drinfel’d associator Φ, ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET, Theorem 4] gives an explicit formula for a Kashiwara-Vergne solution F as follows^{21}:

F = (Φ^{−1}(x, −x − y), e^{−(x+y)/2}Φ(−x − y, y)e^{y/2}).  (13) eq:AETEx

In other words,

F (x) = (Φ^{−1}(x, −x − y))^{−1}xΦ^{−1}(x, −x − y),

and similarly

F (y) = (e^{−(x+y)/2}Φ(−x − y, y)e^{y/2})^{−1}ye^{−(x+y)/2}Φ(−x − y, y)e^{y/2}.

21There are some notational differences between \underline{AlekseevTorossian:KashiwaraVergne} [AT] and \underline{AlekseevEnriquezTorossian:ExplicitSolutions} [AET]. There are sign differences between the formula (\underline{eq:AETEx} \underline{eq:AETEx} 13) and \underline{AlekseevEnriquezTorossian:ExplicitSolutions} [AET] due to notational misalignment, for example our Φ is \underline{AlekseevEnriquezTorossian:ExplicitSolutions} [AET]’s Φ^{−1}. Our notation is consistent with the other papers in this series and the formulas are computationally verified in \underline{Bar-Natan:WKO4} [WKO4].

DRAFT

\xymatrix{Û(𝔩𝔦𝔢_{2}^{2}) \ar[dd]^{L} & Û(𝔩𝔦𝔢_{2}^{2})_{exp} \ar@{^{(}->}[l] \ar[dd]^{L} \ar@{-->}[dr]^{Θ} \\  &  & _{TAut_{2}} \\ A^{sw}(↑_{2}) & A^{sw}(↑_{2})_{exp} \ar@{^{(}->}[l] \ar@{-->}[ur]^{Δ}}

Figure 38. The connection between A^{sw}(↑_{2}) and TAut_{2}.  fig:ATInterpretation

The point of this Appendix is to show that this formula agrees with our Part (2) of the Main Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1, via the correspondences between Drinfel’d associators and expansions Z^{u} ^{\underline{Bar-Natan:GT1}} [BN1, Proposition 3.4] and between KV-solutions and expansions Z^{w} ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Theorem 4.9].

We first relate A^{sw}(↑_{2}) (where V lives) to TAut_{2} (where F lives). A Lie word in x and y can be represented by a binary tree with cyclically oriented vertices, and edges oriented towards a single “head”: the leaves (tails) are labeled by the letters x and y and the vertices represent brackets. For details see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Theorem 3.16] and the discussion following it. There is a tree attaching map l : 𝔱𝔡𝔢𝔯_{2} → P^{sw}(↑_{2}), where P^{sw} denotes the primitive elements of A^{sw}, as follows. Given d ∈ 𝔱𝔡𝔢𝔯_{2}, we have η(d) = (a_{1}, a_{2}) ∈ (𝔩𝔦𝔢_{2})^{2}. Represent the components of η(d) by binary trees, and label the heads with x for Lie words coming from a_{1}, and y for a_{2}. Then, attach all x-labeled leaves to strand 1, y-labeled leaves to strand 2, and the head of each tree below all tails. The order of tails is irrelevant by the TC relation.

Conversely, elements of P^{sw}(↑_{2}) act as tangential derivations on 𝔩𝔦𝔢_{2} ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Proposition 3.19]. The action is by adding a third strand and embedding 𝔩𝔦𝔢_{2} ↪ A(↑_{2}) as x ↦ a_{13} and y ↦ a_{23}. Then P^{sw}(↑_{2}) embedded in P^{sw}(↑_{3}) on the first two strands acts on 𝔩𝔦𝔢_{2} by the adjoint action (taking commutators). Wheels act trivially, and thus one obtains a homomorphism δ : P^{tree}(↑_{2}) → 𝔱𝔡𝔢𝔯_{2}, whose kernel consists of short arrows on either strand. The map δ is a one-sided inverse to l, namely, δ ∘ l = Id_{𝔱𝔡𝔢𝔯_{2}}.

Extending δ to exponentials gives a group homomorphism Δ : A^{w}(↑_{2})_{exp} → TAut_{2}, where A^{w}(↑_{2})_{exp} denotes the group-like part of A^{sw}. For D ∈ A^{w}(↑_{2})_{exp}, the map Δ can be described diagrammatically as follows. Add an extra (third) strand, and embed 𝔩𝔦𝔢_{2} ↪ A(↑_{2}) as before. Then D, embedded in A(↑_{3}) on the first two strands, acts on 𝔩𝔦𝔢_{2} by conjugation; this defines Δ(D).

We construct a map L which completes a commutative triangle as in Figure ^{\underline{fig:ATInterpretation}}^{\underline{fig:ATInterpretation}} 38. At the level of primitives, the map l ∘ θ has the property that δ ∘ (l ∘ θ) = θ. Extend this to the (completed) enveloping algebra Û(𝔩𝔦𝔢_{2}^{2}) as follows. An element of Û(𝔩𝔦𝔢_{2}^{2}) is an (infinite) linear combination of products of Lie words. As with l, represent each Lie word as a labeled tree, but then attach the products of these labeled trees to the two strands by attaching all heads below all tails. The order of tails doesn’t matter, the order of heads is in the order in which the words are multiplied. Call this map L, and note that L is not an algebra homomorphism: it does not respect multiplication in Û(𝔩𝔦𝔢_{2}^{2}). However, the restriction of L to the group-like part Û(𝔩𝔦𝔢_{2}^{2})_{exp}, also denoted L fits into a commutative triangle Θ = Δ∘L. Note that L ≠ e^{l}.

Let Z^{u} be an expansion of sKTG given by the Drinfel’d associator Φ. Let V be the Z^{w}- value of the vertex for the unique homomorphic expansion Z^{w} compatible with Z^{u}. Then

DRAFT

= χ e^{a/2}  e^{a/2}  I  II I  II I  II  φ = ψ  e^{a/2}  ^{Tails}^{Heads} Φ(a_{23}, a_{43})  χ_{1} χ_{2} χ_{3} ψ_{1} ψ_{2}  ψ_{4} ψ_{3}  χ_{3} ψ_{4} ψ_{3} χ_{1} ψ_{2} ψ_{1}  χ_{2}^{Φ^{−1}(a_{2(13)}, −a_{2(13)}) − a_{4(13))}}

Figure 39. To compute φ(Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}) · e^{a_{23}/2} · Φ(a_{23}, a_{43})) we switch to a placement notation in which we mark on each skeleton strand the elements that have arrows ending on it. For this purpose we denote Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}) =: ψ and

Φ(a_{23}, a_{43}) =: χ.  fig:ValueV

π(V)^{−1} π(V)  x
Figure 40. The action of π(V) on the generator x of 𝔩𝔦𝔢_{2}.  fig:ActOnx

F = Δ(V) is a solution to the Kashiwara-Vergne problem ^{\underline{Bar-NatanDancso:WKO2}} [WKO2]. Our goal is to relate the Θ-image of the ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET] formula Θ(Φ^{−1}(x, −x − y), e^{−(x+y)/2}Φ(−x − y, y)e^{y/2}) to Δ(V). Note that wheels are in the kernel of Δ and the ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET] formula only concerns the tree component V^{tree}.

The accomplish this, we need to compute V^{tree} = φ(Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}) · e^{a_{23}/2} · Φ(a_{23}, a_{43})) more explicitly, as shown in Figure ^{\underline{fig:ValueV}}^{\underline{fig:ValueV}} 39, and explained in the caption. The first strand of A^{sw}(↑_{2}) joins strands 1 and 2 in a vertex, and the second strand of A^{w}(↑_{2}) joins strands 3 and 4. Strands 1 and 3 are punctured and strands 2 and 4 are capped. Let us call the two strands of A^{w}(↑_{2}) strand I and strand II to avoid confusion. Recall from the construction of φ that one first slides arrow tails from the capped strands “up” through the vertices, then slides all the heads up from the punctured strands 1 and 3. Thus one obtains an element of A^{w}(↑_{2}) in which all arrow heads are below all tails on both strands.

Now we are ready to compute how π(V) ∈ A^{w}(↑_{2}) acts on the generator x of 𝔩𝔦𝔢_{2} and match this to the formula ^{\underline{eq:AETEx}}^{\underline{eq:AETEx}} 13. Recall the value of π(V) shown in Figure ^{\underline{fig:ValueV}}^{\underline{fig:ValueV}} 39. The generator x is represented by an arrow from the first strand to the added third strand, and the action is by conjugation, as shown in Figure ^{\underline{fig:ActOnx}}^{\underline{fig:ActOnx}} 40. To compute this, one commutes the tail of x to the top of the strand across π(V) using \overrightarrow{ST U} relations, thereby π(V) and π(V)^{−1} cancel, and the result of the action remains. Observe that due to the T C relation, only arrows with heads on strand I act nontrivially on x, in other words only ψ_{1} matters, which came from Φ^{−1}(a_{2(13),−a_{2(13)}−a_{4(13)}}). The arrows a_{23} and a_{43} act trivially on x, so, more simply stated, the action on x is by φ(Φ^{−1}(a_{21}, −a_{21} − a_{41})). Note that L(Φ^{−1}(x, −x − y), 0) = φ(Φ^{−1}(a_{21}, −a_{21} − a_{41})), so Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1 agrees with Formula (^{\underline{eq:AETEx}}^{\underline{eq:AETEx}} 13) in the first component.

DRAFT

Φ_{(24)13}^{−1} Φ_{241} R_{14}^{−1} Φ_{214}^{−1} Φ_{(21)43} R_{(13)(24)} R_{12}^{−1}R_{34}^{−1}^{1}^{2}  ^{3}^{4}
Figure 41. A different expression of β^{b}.  fig:BuckleBraid2

One can proceed similarly for the second component: the action on y is by

φ(Φ^{−1}(a_{23}, −a_{23} − a_{43})e^{a_{23}}Φ(a_{23}, a_{43})) = L(0, Φ^{−1}(x, −x − y)e^{x/2}Φ(x, y)).

While this does not match the second component of Formula (^{\underline{eq:AETEx}}^{\underline{eq:AETEx}} 13), it only differs from it by a hexagon relation. Alternatively, note that one can obtain the second component of the Formula (^{\underline{eq:AETEx}}^{\underline{eq:AETEx}} 13) “on the nose” by starting from an equivalent (isotopic) expression^{22} of β^{b}, as shown in Figure ^{\underline{fig:BuckleBraid2}}^{\underline{fig:BuckleBraid2}} 41. This completes the proof. □

A sequel paper ^{\underline{Bar-Natan:WKO4}} [WKO4] verifies the results in this appendix by explicit computations in low degrees.

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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

Mathematical Sciences Institute, Australian National University, John Dedman Bldg 26, Acton ACT 2601, Australia

Email address: zsuzsanna.dancso@anu.edu.au
URL: http://www.math.toronto.edu/zsuzsi