FINITE TYPE INVARIANTS OF W-KNOTTED OBJECTS I: W-KNOTS AND THE ALEXANDER POLYNOMIAL

DROR BAR-NATAN AND ZSUZSANNA DANCSO

Abstract. w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is \underline{w}ider but \underline{w}eaker than their “\underline{u}sual” counterparts. To get (say) w-knots from u-knots, one has to allow non-planar “virtual” knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the “overcrossings commute” relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.

The group of w-braids was studied (under the name “\underline{w}elded braids”) by Fenn, Rimanyi and Rourke [FRR] and was shown to be isomorphic to the McCool group [Mc] of “basisconjugating” automorphisms of a free group F_{n} — the smallest subgroup of Aut(F_{n}) that contains both braids and permutations. Brendle and Hatcher [BH], in work that traces back to Goldsmith [Gol], have shown this group to be a group of movies of flying rings in ℝ^{3}. Satoh [Sa] studied several classes of w-knotted objects (under the name “\underline{w}eakly-virtual”) and has shown them to be closely related to certain classes of knotted surfaces in ℝ^{4}. So w-knotted objects are algebraically and topologically interesting.

In this article we study finite type invariants of several classes of w-knotted objects. Following Berceanu and Papadima [BP], we construct homomorphic universal finite type invariants of w-braids and of w-tangles. We find that the universal finite type invariant of w-knots is more or less the Alexander polynomial (details inside).

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

The true value of w-knots, though, is likely to emerge later, for we expect them to serve as a \underline{w}armup example for what we expect will be even more interesting — the study of \underline{v}irtual knots, or v-knots. We expect v-knotted objects to provide the global context whose projectivization (or “associated graded structure”) will be the Etingof-Kazhdan theory of deformation quantization of Lie bialgebras [EK].

Date: first edition Sep. 27, 2013, this edition February 24, 2014. The arXiv:1309.7155 edition may be older.

1991 Mathematics Subject Classification. 57M25. Key words and phrases. virtual knots, w-braids, w-knots, w-tangles, knotted graphs, finite type invari-

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

This work was partially supported by NSERC grant RGPIN 262178. Electronic version, videos (wClips)

and related files at [BND2], http://www.math.toronto.edu/~drorbn/papers/WKO/.

Contents

1. Introduction  2
1.1. Dreams  2
1.2. Stories  3
1.3. The Bigger Picture  4
1.4. Plans  5
1.5. Acknowledgement  5
2. w-Braids  6
2.1. Preliminary: Virtual Braids, or v-Braids.  6
2.2. On to w-Braids  8
2.3. Finite Type Invariants of v-Braids and w-Braids  12
2.4. Expansions for w-Braids  15
2.5. Some Further Comments  17
3. w-Knots  22
3.1. v-Knots and w-Knots  22
3.2. Finite Type Invariants of v-Knots and w-Knots  26
3.3. Some Dimensions  28
3.4. Expansions for w-Knots  29
3.5. Jacobi Diagrams, Trees and Wheels  30
3.6. The Relation with Lie Algebras  35
3.7. The Alexander Polynomial  37
3.8. Proof of Theorem 3.27  38
3.9. The Relationship with u-Knots  45
4. Odds and Ends  47
4.1. What means “closed form”?  47
4.2. Arrow Diagrams to Degree 2  47
5. Glossary of notation  49
References  51

1. Introduction

1.1. Dreams. We have a dream^{1}, at least partially founded on reality, that many of the difficult algebraic equations in mathematics, especially those that are written in graded spaces, more especially those that are related in one way or another to quantum groups [Dr1], and even more especially those related to the work of Etingof and Kazhdan [EK], can be understood, and indeed, would appear more natural, in terms of finite type invariants of various topological objects.

We believe this is the case for Drinfel’d’s theory of associators [Dr2], which can be interpreted as a theory of well-behaved universal finite type invariants of parenthesized tangles^{2} [LM2, BN3], and even more elegantly, as a theory of universal finite type invariants of knotted trivalent graphs [Da].

1Understanding the authors’ history and psychology ought never be necessary to understand their papers, yet it may be helpful. Nothing material in the rest of this paper relies on Section 1.1.

2“q-tangles” in [LM2], “non-associative tangles” in [BN3].

We believe this is the case for Drinfel’d’s “Grothendieck-Teichmuller group” [Dr3] which is better understood as a group of automorphisms of a certain algebraic structure, also related to universal finite type invariants of parenthesized tangles [BN6].

And we’re optimistic, indeed we believe, that sooner or later the work of Etingof and Kazhdan [EK] on quantization of Lie bialgebras will be re-interpreted as a construction of a well-behaved universal finite type invariant of virtual knots [Ka2] or of some other class of virtually knotted objects. Some steps in that direction were taken by Haviv [Hav].

We have another dream, to construct a useful “Algebraic Knot Theory”. As at least a partial writeup exists [BN8], we’ll only state that an important ingredient necessary to fulfil that dream would be a “closed form”^{3} formula for an associator, at least in some reduced sense. Formulae for associators or reduced associators were in themselves the goal of several studies undertaken for various other reasons [LM1, Lie, Kur, Lee1].

1.2. Stories. Thus the first named author, DBN, was absolutely delighted when in January 2008 Anton Alekseev described to him his joint work [AT] with Charles Torossian — Anton told DBN that they found a relationship between the Kashiwara-Vergne conjecture [KV], a cousin of the Duflo isomorphism (which DBN already knew to be knot-theoretic [BLT]), and associators taking values in a space called 𝔰𝔡𝔢𝔯, which he could identify as “tree-level Jacobi diagrams”, also a knot-theoretic space related to the Milnor invariants [BN2, HM]. What’s more, Anton told DBN that in certain quotient spaces the Kashiwara-Vergne conjecture can be solved explicitly; this should lead to some explicit associators!

So DBN spent the following several months trying to understand [AT], and this paper is a summary of these efforts. The main thing we learned is that the Alekseev-Torossian paper, and with it the Kashiwara-Vergne conjecture, fit very nicely with our first dream recalled above, about interpreting algebra in terms of knot theory. Indeed much of [AT] can be reformulated as a construction and a discussion of a well-behaved universal finite type invariant Z of a certain class of knotted objects (which we will call here w-knotted), a certain natural quotient of the space of virtual knots (more precisely, virtual trivalent tangles). And our hopes remain high that later we (or somebody else) will be able to exploit this relationship in directions compatible with our second dream recalled above, on the construction of an “algebraic knot theory”.

The story, in fact, is prettier than we were hoping for, for it has the following additional qualities:

• w-Knotted objects are quite interesting in themselves: as stated in the abstract, they are related to combinatorial group theory via “basis-conjugating” automorphisms of a free group F_{n}, to groups of movies of flying rings in ℝ^{3}, and more generally, to certain classes of knotted surfaces in ℝ^{4}. The references include [BH, FRR, Gol, Mc, Sa].

• The “chord diagrams” for w-knotted objects (really, these are “arrow diagrams”) describe formulae for invariant tensors in spaces pertaining to not-necessarily-metrized Lie algebras in much of the same way as ordinary chord diagrams for ordinary knotted objects describe formulae for invariant tensors in spaces pertaining to metrized Lie algebras. This observation is bound to have further implications. v-Knots w-Knots

3The phrase “closed form” in itself requires an explanation. See Section 4.1.

u-Knots

Ordinary (\underline{u}sual) knotted objects in 3D — braids, knots, links, tangles, knotted graphs, etc.

\underline{V}irtual knotted objects — “algebraic” knotted objects, or “not specifically embedded” knotted objects; knots drawn on a surface, modulo stabilization.

Ribbon knotted objects in 4D; “flying rings”. Like v, but also with “overcrossings commute”.

Chord diagrams and Jacobi diagrams, modulo 4T, ST U, IHX, etc.

Arrow diagrams and v-Jacobi diagrams, modulo 6T and various “directed” ST Us and IHXs, etc.

Like v, but also with “tails commute”. Only “two in one out” internal vertices.

Finite dimensional metrized Lie algebras, representations, and associated spaces.

Finite dimensional Lie bi-algebras, representations, and associated spaces.

Finite dimensional cocommutative Lie bi-algebras (i.e., 𝔤⋉𝔤^{∗}), representations, and associated spaces.

The Drinfel’d theory of associators.

Likely, quantum groups and the Etingof-Kazhdan theory of quantization of Lie bialgebras.

The Kashiwara-Vergne-Alekseev-Torossian theory of convolutions on Lie groups and Lie algebras.

Combinatorics

Low Algebra

High Algebra Topology

Figure 1. The u-v-w Stories

• Arrow diagrams also describe the Feynman diagrams of topological BF theory [CCM, CCFM] and of a certain class of Chern-Simons theories [Na]. Thus it is likely that our story is directly related to quantum field theory^{4}.

• When composed with the map from knots to w-knots, Z becomes the Alexander polynomial. For links, it becomes an invariant stronger than the multi-variable Alexander polynomial which contains the multi-variable Alexander polynomial as an easily identifiable reduction. On other w-knotted objects Z has easily identifiable reductions that can be considered as “Alexander polynomials” with good behaviour relative to various knot-theoretic operations — cablings, compositions of tangles, etc. There is also a certain specific reduction of Z that can be considered as the “ultimate Alexander polynomial” — in the appropriate sense, it is the minimal extension of the Alexander polynomial to other knotted objects which is well behaved under a whole slew of knot theoretic operations, including the ones named above.

1.3. The Bigger Picture. Parallel to the w-story run the possibly more significant u-story and v-story. The u-story is about u-knots, or more generally, u-knotted objects (braids, links, tangles, etc.), where “u” stands for \underline{u}sual; hence the u-story is about ordinary knot theory. The v-story is about v-knots, or more generally, v-knotted objects, where “v” stands for \underline{v}irtual, in the sense of Kauffman [Ka2].

4Some non-perturbative relations between BF theory and w-knots was discussed by Baez, Wise and Crans [BWC].

The three stories, u, v, and w, are different from each other. Yet they can be told along similar lines — first the knots (topology), then their finite type invariants and their “chord diagrams” (combinatorics), then those map into certain universal enveloping algebras and similar spaces associated with various classes of Lie algebras (low algebra), and finally, in order to construct a “good” universal finite type invariant, in each case one has to confront a certain deeper algebraic subject (high algebra). These stories are summarized in a table form in Figure 1.

u-Knots map into v-knots, and v-knots map into w-knots^{5}. The other parts of our stories, the “combinatorics” and “low algebra” and “high algebra” rows of Figure 1, are likewise related, and this relationship is a crucial part of our overall theme. Thus we cannot and will not tell the w-story in isolation, and while it is central to this article, we will necessarily also include some episodes from the u and v series.

1.4. Plans. Our order of proceedings is: w-braids, w-knots, generalities, w-tangles, wtangled foams. For more detailed information consult the “Section Summary” paragraphs below and at the beginning of each of the sections. An “odds and ends” section follows on page 47, and a glossary of notation is on page 49.

Section 2, w-Braids. (page 6) This section is largely a compilation of existing literature, though we also introduce the language of arrow diagrams that we use throughout the rest of the paper. In 2.1 and 2.2 we define vbraids and then w-braids and survey their relationship with basis-conjugating automorphisms of free groups and with “the group of (horizontal) flying rings in ℝ^{3}” (really, a group of knotted tubes in ℝ^{4}). In 2.3 we play the usual game of introducing finite type invariants, weight systems, chord diagrams (arrow diagrams, for this case), and 4T-like relations. In 2.4 we define and construct a universal finite type invariant Z for w-braids — it turns out that the only algebraic tool we need to use is the formal exponential function exp(a) := ∑ a^{n}/n!. In 2.5 we study some good algebraic properties of Z, its injectivity, and its uniqueness, and we conclude with the slight modifications needed for the study of non-horizontal flying rings.

Section 3, w-Knots. (page 22) In 3.1 we define v-knots and w-knots (long v-knots and long w-knots, to be precise) and discuss the map v → w. In 3.2 we determine the space of “chord diagrams” for w-knots to be the space A^{w}(↑) of arrow diagrams modulo \overrightarrow{4T} and TC relations and in 3.3 we compute some relevant dimensions. In 3.5 we show that A^{w}(↑) can be re-interpreted as a space of trivalent graphs modulo STU- and IHX-like relations, and is therefore related to Lie algebras (Sec. 3.6). This allows us to completely determine A^{w}(↑). With no difficulty at all in 3.4 we construct a universal finite type invariant for w-knots. With a bit of further difficulty we show in Sec. 3.7 that it is essentially equal to the Alexander polynomial.

1.5. Acknowledgement. We wish to thank Anton Alekseev, Jana Archibald, Scott Carter, Karene Chu, Iva Halacheva, Joel Kamnitzer, Lou Kauffman, Peter Lee, Louis Leung, Dylan Thurston, Lucy Zhang, and Jean-Baptiste Meilhan for comments and suggestions.

5Though the composition “u → v → w” is not 0. In fact, the composed map u → w is injective.

2. w-Braids

Section Summary. This section is largely a compilation of existing literature, though we also introduce the language of arrow diagrams that we use throughout the rest of the paper. In 2.1 and 2.2 we define v-braids and then w-braids and survey their relationship with basis-conjugating automorphisms of free groups and with “the group of (horizontal) flying rings in ℝ^{3}” (really, a group of knotted tubes in ℝ^{4}). In 2.3 we play the usual game of introducing finite type invariants, weight systems, chord diagrams (arrow diagrams, for this case), and 4T-like relations. In 2.4 we define and construct a universal finite type invariant Z for w-braids — it turns out that the only algebraic tool we need to use is the formal exponential function exp(a) := ∑ a^{n}/n!. In 2.5 we study some good algebraic properties of Z, its injectivity, and its uniqueness, and we conclude with the slight modifications needed for the study of non-horizontal flying rings.

2.1. Preliminary: Virtual Braids, or v-Braids. Our main object of study for this section, w-braids, are best viewed as “virtual braids” [Ba, KL, BB], or v-braids, modulo one additional relation. Hence we start with v-braids.

It is simplest to define v-braids in terms of generators and relations, either algebraically or pictorially. This can be done in at least two ways — the easier-at-first but philosophicallyless-satisfactory “planar” way, and the harder to digest but morally more correct “abstract” way.^{6}

2.1.1. The “Planar” Way. For a natural number n set vB_{n} to be the group generated by symbols σ_{i} (1 ≤ i ≤ n−1), called “crossings” and graphically represented by an overcrossing � “between strand i and strand i + 1” (with inverse �)^{7}, and s_{i}, called “virtual crossings” and graphically represented by a non-crossing, �, also “between strand i and strand i + 1”, subject to the following relations:

• The subgroup of vB_{n} generated by the virtual crossings s_{i} is the symmetric group S_{n}, and the s_{i}’s correspond to the transpositions (i, i + 1). That is, we have

s_{i}^{2} = 1,  s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1},  and if |i − j| > 1 then s_{i}s_{j} = s_{j}s_{i}.  (1)

In pictures, this is

i  ^{i+2}^{i+1}  ^{i}  ^{i+2}^{i+1}^{i} ^{i+1} ^{i} ^{i+1}  ^{i} ^{i+1}  ^{i} ^{i+1}  ^{j} ^{j+1}^{j} ^{j+1}^{=}  ^{=}  ^{=}  (2)

Note that we read our braids from bottom to top.

• The subgroup of vB_{n} generated by the crossings σ_{i}’s is the usual braid group uB_{n}, and σ_{i} corresponds to the braiding of strand i over strand i + 1. That is, we have

σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1},  and if |i − j| > 1 then σ_{i}σ_{j} = σ_{j}σ_{i}.  (3)

6Compare with a similar choice that exists in the definition of manifolds, as either appropriate subsets of some ambient Euclidean spaces (module some equivalences) or as abstract gluings of coordinate patches (modulo some other equivalences). Here in the “planar” approach of Section 2.1.1 we consider v-braids as “planar” objects, and in the “abstract approach” of Section 2.1.2 they are just “gluings” of abstract “crossings”, not drawn anywhere in particular.

7We sometimes refer to � as a “positive crossing” and to � as a “negative crossing”.

In pictures, dropping the indices, this is

{= \atop =}  (4)

The first of these relations is the “Reidemeister 3 move”^{8} of knot theory. The second is sometimes called “locality in space” [BN3].

• Some “mixed relations”,

s_{i}σ_{i+1}^{±1}s_{i} = s_{i+1}σ_{i}^{±1}s_{i+1},  and if |i − j| > 1 then s_{i}σ_{j} = σ_{j}s_{i}.  (5)

In pictures, this is

=  ,  =  and  =  (6)

Remark 2.1. The “skeleton” of a v-braid B is the set of strands appearing in it, retaining the association between their beginning and ends but ignoring all the crossing information. More precisely, it is the permutation induced by tracing along B, and even more precisely it is the image of B via the “skeleton morphism” ς: vB_{n} → S_{n} defined by ς(σ_{i}) = ς(s_{i}) = s_{i} (or pictorially, by ς(�) = ς(�) = �). Thus the symmetric group S_{n} is both a subgroup and a quotient group of vB_{n}.

Like there are pure braids to accompany braids, there are pure virtual braids as well:

Definition 2.2. A pure v-braid is a v-braid whose skeleton is the identity permutation; the group PvB_{n} of all pure v-braids is simply the kernel of the skeleton morphism ς: vB_{n} → S_{n}.

We note the sequence of group homomorphisms

1 ⟶ PvB_{n} ⟶ vB_{n} ⟶^{ς} S_{n} ⟶ 1.  (7)

This sequence is exact and split, with the splitting given by the inclusion S_{n} ↪ vB_{n} mentioned above (1). Therefore we have that

vB_{n} = PvB_{n} ⋊ S_{n}.  (8)

2.1.2. The “Abstract” Way. The relations (2) and (6) that govern the behaviour of virtual crossings precisely say that virtual crossings really are “virtual” — if a piece of strand is routed within a braid so that there are only virtual crossings around it, it can be rerouted in any other “virtual only” way, provided the ends remain fixed (this is Kauffman’s “detour move” [Ka2, KL]). Since a v-braid B is independent of the routing of virtual pieces of strand, we may as well never supply this routing information.

8The Reidemeister 2 move is the relations σ_{i}σ_{i}^{−1} = 1 which is part of the definition of “a group”. There is no Reidemeister 1 move in the theory of braids.

1 2 3

Thus for example, a perfectly fair verbal description of the (pure!) v-braid on the right is “strand 1 goes over strand 3 by a positive crossing then likewise positively over strand 2 then negatively over 3 then 2 goes positively over 1”. We don’t need to specify how strand 1 got to be near strand 3 so it can go over it — it got there by means of virtual crossings, and it doesn’t matter how. Hence we arrive at the following “abstract” presentation of PvB_{n} and vB_{n}:

Proposition 2.3. (E.g. [Ba])

(1) The group PvB_{n} of pure v-braids is isomorphic to the group generated by symbols σ_{ij} for 1 ≤ i ≠ j ≤ n (meaning “strand i crosses over strand j at a positive crossing”^{9}), subject to the third Reidemeister move and to locality in space (compare with (3) and (4)):

σ_{ij}σ_{ik}σ_{jk} = σ_{jk}σ_{ik}σ_{ij}  whenever  |{i, j, k}| = 3,
σ_{ij}σ_{kl} = σ_{kl}σ_{ij}  whenever  |{i, j, k, l}| = 4.

(2) If τ ∈ S_{n}, then with the action σ_{ij}^{τ} := σ_{τi,τj} we recover the semi-direct product decomposition vB_{n} = PvB_{n} ⋊ S_{n}. □

2.2. On to w-Braids. To define w-braids, we break the symmetry between over crossings and under crossings by imposing one of the “forbidden moves” virtual knot theory, but not the other:

σ_{i}σ_{i+1}s_{i} = s_{i+1}σ_{i}σ_{i+1},  yet s_{i}σ_{i+1}σ_{i} ≠ σ_{i+1}σ_{i}s_{i+1}.  (9)

Alternatively,

σ_{ij}σ_{ik} = σ_{ik}σ_{ij}, yet σ_{ik}σ_{jk} ≠ σ_{jk}σ_{ik}.

In pictures, this is

i ^{j} ^{k}^{=}^{i} ^{j} ^{k}  ^{i} ^{j} ^{k}^{≠}^{i} ^{j} ^{k}  (10)

The relation we have just imposed may be called the “unforbidden relation”, or, perhaps more appropriately, the “overcrossings commute” relation (OC). Ignoring the non-crossings^{10} �, the OC relation says that it is the same if strand i first crosses over strand j and then over strand k, or if it first crosses over strand k and then over strand j. The “undercrossings commute” relation UC, the one we do not impose in (9), would say the same except with “under” replacing “over”.

Definition 2.4. The group of w-braids is wB_{n} := vB_{n}/OC. Note that ς descends to wB_{n} and hence we can define the group of pure w-braids to be PwB_{n} := ker ς: wB_{n} → S_{n}. We still have a split exact sequence as at (7) and a semi-direct product decomposition wB_{n} = PwB_{n} ⋊ S_{n}.

9The inverse, σ_{ij}^{−1}, is “strand i crosses over strand j at a negative crossing”

10Why this is appropriate was explained in the previous section.

Exercise 2.5. Show that the OC relation is equivalent to the relation

σ_{i}^{−1}s_{i+1}σ_{i} = σ_{i+1}s_{i}σ_{i+1}^{−1} or _{=}

While mostly in this paper the pictorial / algebraic definition of w-braids (and other wknotted objects) will suffice, we ought describe at least briefly 2-3 further interpretations of wB_{n}:

2.2.1. The group of flying rings. Let X_{n} be the space of all placements of n numbered disjoint geometric circles in ℝ^{3}, such that all circles are parallel to the xy plane. Such placements will be called horizontal^{11}. A horizontal placement is determined by the centres in ℝ^{3} of the n circles and by n radii, so dim X_{n} = 3n + n = 4n. The permutation group S_{n} acts on X_{n} by permuting the circles, and one may think of the quotient X̃_{n} := X_{n}/S_{n} as the space of all horizontal placements of n unmarked circles in ℝ^{3}. The fundamental group π_{1}(X̃_{n}) is a group of paths traced by n disjoint horizontal circles (modulo homotopy), so it is fair to think of it as “the group of flying rings”.

Theorem 2.6. The group of pure w-braids PwB_{n} is isomorphic to the group of flying rings π_{1}(X_{n}). The group wB_{n} is isomorphic to the group of unmarked flying rings π_{1}(X̃_{n}).

For the proof of this theorem, see [Gol, Sa] and especially [BH]. Here we will contend ourselves with pictures describing the images of the generators of wB_{n} in π_{1}(X̃_{n}) and a few comments:

σ_{i} = s_{i} =^{i} ^{i + 1}  ^{i} ^{i + 1}

Thus we map the permutation s_{i} to the movie clip in which ring number i trades its place with ring number i + 1 by having the two flying around each other. This acrobatic feat is performed in ℝ^{3} and it does not matter if ring number i goes “above” or “below” or “left” or “right” of ring number i + 1 when they trade places, as all of these possibilities are homotopic. More interestingly, we map the braiding σ_{i} to the movie clip in which ring i + 1 shrinks a bit and flies through ring i. It is a worthwhile exercise for the reader to verify that the relations in the definition of wB_{n} become homotopies of movie clips. Of these relations it is most interesting to see why the “overcrossings commute” relation σ_{i}σ_{i+1}s_{i} = s_{i+1}σ_{i}σ_{i+1} holds, yet the “undercrossings commute” relation σ_{i}^{−1}σ_{i+1}^{−1}s_{i} = s_{i+1}σ_{i}^{−1}σ_{i+1}^{−1} doesn’t.

Exercise 2.7. To be perfectly precise, we have to specify the fly-through direction. In our notation, σ_{i} means that the ring corresponding to the under-strand approaches the bigger ring representing the over-strand from below, flies through it and exists above. For σ_{i}^{−1} we are “playing the movie backwards”, i.e., the ring of the under-strand comes from above and exits below the ring of the over-strand.

11 For the group of non-horizontal flying rings see Section 2.5.4

Let “the signed w braid group”, swB_{n}, be the group of horizontal flying rings where both fly-through directions are allowed. This introduces a “sign” for each crossing σ_{i}:

i ^{i + 1}  ^{i} ^{i + 1}^{+}  ^{−} σ_{i−} = σ_{i+} =

In other words, swB_{n} is generated by s_{i}, σ_{i+} and σ_{i−}, for i = 1, ..., n. Check that in swB_{n} σ_{i−} = s_{i}σ_{i+}^{−1}s_{i}, and this, along with the other obvious relations implies swB_{n} ≅ wB_{n}.

For a rigorous discussion of orientations and signs, see Section ??.

2.2.2. Certain ribbon tubes in ℝ^{4}. With time as the added dimension, a flying ring in ℝ^{3} traces a tube (an annulus) in ℝ^{4}, as shown in the picture below:

s_{i} = ^{i} ^{i + 1}  σ_{i} = ^{i} ^{i + 1}

Note that we adopt here the drawing conventions of Carter and Saito [CS] — we draw surfaces as if they were projected from ℝ^{4} to ℝ^{3}, and we cut them open whenever they are “hidden” by something with a higher fourth coordinate.

Note also that the tubes we get in ℝ^{4} always bound natural 3D “solids” — their “insides”, in the pictures above. These solids are disjoint in the case of s_{i} and have a very specific kind of intersection in the case of σ_{i} — these are transverse intersections with no triple points, and their inverse images are a meridional disk on the “thin” solid tube and an interior disk on the “thick” one. By analogy with the case of ribbon knots and ribbon singularities in ℝ^{3} (e.g. [Ka1, Chapter V]) and following Satoh [Sa], we call this kind if intersections of solids in ℝ^{4} “ribbon singularities” and thus our tubes in ℝ^{4} are always “ribbon tubes”.

2.2.3. Basis conjugating automorphisms of F_{n}. Let F_{n} be the free (non-Abelian) group with generators ξ_{1}, …, ξ_{n}. Artin’s theorem (Theorems 15 and 16 of [Ar]) says that the (usual) braid group uB_{n} (equivalently, the subgroup of wB_{n} generated by the σ_{i}’s) has a faithful right action on F_{n}. In other words, uB_{n} is isomorphic to a subgroup H of Aut^{op}(F_{n}) (the group of automorphisms of F_{n} with opposite multiplication; ψ_{1}ψ_{2} := ψ_{2} ∘ ψ_{1}). Precisely, using (ξ, B) ↦ ξ⫽B to denote the right action of Aut^{op}(F_{n}) on F_{n}, the subgroup H consists of those automorphisms B: F_{n} → F_{n} of F_{n} that satisfy the following two conditions:

(1) B maps any generator ξ_{i} to a conjugate of a generator (possibly different). That is, there is a permutation β ∈ S_{n} and elements a_{i} ∈ F_{n} so that for every i,

ξ_{i} ⫽ B = a_{i}^{−1}ξ_{βi}a_{i}.  (11)

(2) B fixes the ordered product of the generators of F_{n},

ξ_{1}ξ_{2} ⋯ ξ_{n} ⫽ B = ξ_{1}ξ_{2} ⋯ ξ_{n}.

McCool’s theorem [Mc] says that the same holds true^{12} if one replaces the braid group uB_{n} with the bigger group wB_{n} and drops the second condition above. So wB_{n} is precisely the group of “basis-conjugating” automorphisms of the free group F_{n}, the group of those automorphisms which map any “basis element” in {ξ_{1}, …, ξ_{n}} to a conjugate of a (possibly different) basis element.

The relevant action is explicitly defined on the generators of wB_{n} and F_{n} as follows (with the omitted generators of F_{n} always fixed):

(ξ_{i}, ξ_{i+1}) ⫽ s_{i} = (ξ_{i+1}, ξ_{i}) (ξ_{i}, ξ_{i+1}) ⫽ σ_{i} = (ξ_{i+1}, ξ_{i+1}ξ_{i}ξ_{i+1}^{−1}) ξ_{j} ⫽ σ_{ij} = ξ_{i}ξ_{j}ξ_{i}^{−1} (12)

It is a worthwhile exercise to verify that ⫽ respects the relations in the definition of wB_{n} and that the permutation β in (11) is the skeleton ς(B).

There is a more conceptual description of ⫽, in terms of the structure of wB_{n+1}. Consider the inclusions

wB_{n} ⟶^{ι} wB_{n+1} ⟵^{i_{u}} F_{n}.  (13)

1 i i+1 n n+1 ⋯  ⋯ ξ_{i} ↦

Here ι is the inclusion of wB_{n} into wB_{n+1} by adding an inert (n + 1)−st strand (it is injective as it has a well defined one sided inverse — the deletion of the (n + 1)-st strand). The inclusion i_{u} of the free group F_{n} into wB_{n+1} is defined by i_{u}(ξ_{i}) := σ_{i,n+1}. The image i_{u}(F_{n}) ⊂ wB_{n+1} is the set of all w-braids whose first n strands are straight and vertical, and whose (n+1)-st strand wanders among the first n strands mostly virtually (i.e., mostly using virtual crossings), occasionally slipping under one of those n strands, but never going over anything. In the “flying rings” picture of Section 2.2.1, the image i_{u}(F_{n}) ⊂ wB_{n+1} can be interpreted as the fundamental group of the complement in ℝ^{3} of n stationary rings (which is indeed F_{n}) — in i_{u}(F_{n}) the only ring in motion is the last, and it only goes under, or “through”, other rings, so it can be replaced by a point object whose path is an element of the fundamental group. The injectivity of i_{u} follows from this geometric picture.

B^{−1} B γ

One may explicitly verify that i_{u}(F_{n}) is normalized by ι(wB_{n}) in wB_{n+1} (that is, the set i_{u}(F_{n}) is preserved by conjugation by elements of ι(wB_{n})). Thus the following definition (also shown as a picture on the right) makes sense, for B ∈ wB_{n} ⊂ wB_{n+1} and for γ ∈ F_{n} ⊂ wB_{n+1}:

γ ⫽ B := i_{u}^{−1}(B^{−1}γB)  (14)

It is a worthwhile exercise to recover the explicit formulae in (12) from the above definition.

Warning 2.8. People familiar with the Artin story for ordinary braids should be warned that even though wB_{n} acts on F_{n} and the action is induced from the inclusions in (13) in much of the same way as the Artin action is induced by inclusions uB_{n} ⟶^{ι} uB_{n+1} ⟵^{i} F_{n}, there are also some differences, and some further warnings apply:

• In the ordinary Artin story, i(F_{n}) is the set of braids in uB_{n+1} whose first n strands are unbraided (that is, whose image in uB_{n} via “dropping the last strand” is the identity). This is not true for w-braids. For w-braids, in i_{u}(F_{n}) the last strand always goes “under” all other strands (or just virtually crosses them), but never over.

12Though see Warning 2.8.

• Thus unlike the isomorphism PuB_{n+1} ≅ PuB_{n} ⋉ F_{n}, it is not true that PwB_{n+1} is isomorphic to PwB_{n} ⋉ F_{n}.

• The Overcrossings Commute relation imposed in wB breaks the symmetry between overcrossings and undercrossings. Thus let i_{o}: F_{n} → wB_{n} be the “opposite” of i_{u}, mapping into braids in which the last strand is always “over” or virtual. Then i_{o} is not injective (its image is in fact Abelian) and its image is not normalized by ι(wB_{n}). So there is no “second” action of wB_{n} on F_{n} defined using i_{o}.

• For v-braids, both i_{u} and i_{o} are injective and there are two actions of vB_{n} on F_{n} — one defined by first projecting into w-braids, and the other defined by first projecting into vbraids modulo “Undercrossings Commute”. Yet v-braids contain more information than these two actions can see. The “Kishino” v-braid below, for example, is visibly trivial if either overcrossings or undercrossings are made to commute, yet by computing its Kauffman bracket we know it is non-trivial as a v-braid [BND2, “The Kishino Braid”]:

a  b  (The commutator ab^{−1}a^{−1}b of v-braids a, b annihilated by OC/UC, respectively, with a minor cancellation.)

Problem 2.9. Is PwB_{n} a semi-direct product of free groups? Note that both PuB_{n} and PvB_{n} are such semi-direct products: For PuB_{n}, this is the well known “combing of braids”; it follows from PuB_{n} ≅ PuB_{n−1} ⋉ F_{n−1} and induction. For PvB_{n}, it is a result stated in [Ba] (though our own understanding of [Ba] is incomplete).

Remark 2.10. Note that Gutiérrez and Krstić [GK] find “normal forms” for the elements of PwB_{n}, yet they do not decide whether PwB_{n} is “automatic” in the sense of [Ep].

2.3. Finite Type Invariants of v-Braids and w-Braids. Just as we had two definitions for v-braids (and thus w-braids) in Section 2.1, we will give two (obviously equivalent) developments of the theory of finite type invariants of v-braids and w-braids — a pictorial/topological version in Section 2.3.1, and a more abstract algebraic version in Section 2.3.2.

2.3.1. Finite Type Invariants, the Pictorial Approach. In the standard theory of finite type invariants of knots (also known as Vassiliev or Goussarov-Vassiliev invariants) [Gou1, Vas, BN1, BN7] one progresses from the definition of finite type via iterated differences to chord diagrams and weight systems, to 4T (and other) relations, to the definition of universal finite type invariants, and beyond. The exact same progression (with different objects playing similar roles, and sometimes, when yet insufficiently studied, with the last step or two missing) is also seen in the theories of finite type invariants of braids [BN5], 3-manifolds [Oh, LMO, Le], virtual knots [GPV, Po] and of several other classes of objects. We thus assume that the reader has familiarity with these basic ideas, and we only indicate briefly how they are implemented in the case of v-braids and w-braids.

Much like the formula � → � − � of the Vassiliev-Goussarov fame, given a v-braid invariant V : vB_{n} → A valued in some Abelian group A, we extend it to “singular” vbraids, braids that contain “semi-virtual crossings” like � and � using the formulae V (�) := V (�)−V (�) and V (�) := V (�)−V (�) (see [GPV, Po, BHLR]). We say that “V is of type m” if its extension vanishes on singular v-braids having more than m semi-virtual crossings. Up to invariants of lower type, an invariant of type m is determined by its “weight system”,

1 2 3 4  1 2 3 4 1 2 3 4 1 2 3 4  β D  i j k l  i j k l

Figure 2. On the left, a 3-singular v-braid and its corresponding 3-arrow diagram. A self-explanatory algebraic notation for this arrow diagram is (a_{12}a_{41}a_{23}, 3421). picture and in algebraic notation. Note that we regard arrow diagrams as graph-theoretic objects, and hence the two arrow diagrams on the right, whose underlying graphs are the same, are regarded as equal. In algebraic notation this means that we always impose the relation a_{ij}a_{kl} = a_{kl}a_{ij} when the indices i, j, k, and l are all distinct.

i j k + i j k + i j k = i j k + i j k + i j k
a_{ij}a_{ik} + a_{ij}a_{jk} + a_{ik}a_{jk} = a_{ik}a_{ij} + a_{jk}a_{ij} + a_{jk}a_{ik}
or [a_{ij}, a_{ik}] + [a_{ij}, a_{jk}] + [a_{ik}, a_{jk}] = 0

Figure 3. The 6T relation. Standard knot theoretic conventions apply — only the relevant parts of each diagram is shown; in reality each diagram may have further vertical strands and horizontal arrows, provided the extras are the same in all 6 diagrams. Also, the vertical strands are in no particular order — other valid 6T relations are obtained when those strands are permuted in other ways.

which is a functional W = W_{m}(V) defined on “m-singular v-braids modulo � = � = �”. Let us denote the vector space of all formal linear combinations of such equivalence classes by G_{m}D_{n}^{v}. Much as m-singular knots modulo � = � can be identified with chord diagrams, the basis elements of G_{m}D_{n}^{v} can be identified with pairs (D, β), where D is a horizontal arrow diagram and β is a “skeleton permutation”. See Figure 2.

We assemble the spaces G_{m}D_{n}^{v} together to form a single graded space, D_{n}^{v} := ⊕_{m=0}^{∞}G_{m}D_{n}^{v}. Note that throughout this paper, whenever we write an infinite direct sum, we automatically complete it. Thus in D_{n}^{v} we allow infinite sums with one term in each homogeneous piece G_{m}D_{n}^{v}.

In the standard finite-type theory for knots, weight systems always satisfy the 4T relation, and are therefore functionals on A := D/4T. Likewise, in the case of v-braids, weight systems satisfy the “6T relation” of [GPV, Po, BHLR], shown in Figure 3, and are therefore functionals on A_{n}^{v} := D_{n}^{v}/6T. In the case of w-braids, the “overcrossings commute” relation (9) implies the “Tails Commute” (TC) relation on the level of arrow diagrams, and in the presence of the TC relation, two of the terms in the 6T relation drop out, and what

\begin{matrix}^{^{i} ^{j} ^{k} ^{=} ^{i} ^{j} ^{k}} & ^{^{i} ^{j} ^{k} ^{+} ^{i} ^{j} ^{k} ^{=} ^{i} ^{j} ^{k} ^{+} ^{i} ^{j} ^{k}} \\ a_{ij}a_{ik} = a_{ik}a_{ij} & a_{ij}a_{jk} + a_{ik}a_{jk} = a_{jk}a_{ij} + a_{jk}a_{ik} \\ or [a_{ij}, a_{ik}] = 0 & or [a_{ij} + a_{ik}, a_{jk}] = 0 \end{matrix}

Figure 4. The TC and the \overrightarrow{4T} relations.

remains is the “\overrightarrow{4T}” relation. These relations are shown in Figure 4. Thus weight systems of finite type invariants of w-braids are linear functionals on A_{n}^{w} := D_{n}^{v}/T C, \overrightarrow{4T}.

The next question that arises is whether we have already found all the relations that weight systems always satisfy. More precisely, given a degree m linear functional on A_{n}^{v} = D_{n}^{v}/6T (or on A_{n}^{w} = D_{n}^{v}/T C, \overrightarrow{4T}), is it always the weight system of some type m invariant V of v-braids (or w-braids)? As in every other theory of finite type invariants, the answer to this question is affirmative if and only if there exists a “universal finite type invariant” (or simply, an “expansion”) of v-braids (w-braids):

Definition 2.11. An expansion for v-braids (w-braids) is an invariant Z: vB_{n} → A_{n}^{v} (or Z: wB_{n} → A_{n}^{w}) satisfying the following “universality condition”:

• If B is an m-singular v-braid (w-braid) and D ∈ G_{m}D_{n}^{v} is its underlying arrow diagram as in Figure 2, then

Z(B) = D + (terms of degree > m).

Indeed if Z is an expansion and W ∈ G_{m}A^{⋆},^{13} the universality condition implies that W ∘ Z is a finite type invariant whose weight system is W. To go the other way, if (D_{i}) is a basis of A consisting of homogeneous elements, if (W_{i}) is the dual basis of A^{⋆} and (V_{i}) are finite type invariants whose weight systems are the W_{i}’s, then Z(B) := ∑_{i} D_{i}V_{i}(B) defines an expansion.

In general, constructing a universal finite type invariant is a hard task. For knots, one uses either the Kontsevich integral or perturbative Chern-Simons theory (also known as “configuration space integrals” [BT] or “tinker-toy towers” [Th]) or the rather fancy algebraic theory of “Drinfel’d associators” (a summary of all those approaches is at [BS]). For homology spheres, this is the “LMO invariant” [LMO, Le] (also the “Århus integral” [BGRT2]). For v-braids, we still don’t know if an expansion exists. As we shall see below, the construction of an expansion for w-braids is quite easy.

2.3.2. Finite Type Invariants, the Algebraic Approach. For any group G, one can form the group algebra 𝔽G for some field 𝔽 by allowing formal linear combinations of group elements and extending multiplication linearly. The augmentation ideal is the ideal generated by differences, or equivalently, the set of linear combinations of group elements whose coefficients sum to zero:

I := {∑_{i=1}^{k} a_{i}g_{i}: a_{i} ∈ 𝔽, g_{i} ∈ G, ∑_{i=1}^{k} a_{i} = 0} .

13A here denotes either A_{n}^{v} or A_{n}^{w}, or in fact, any of many similar spaces that we will discuss later on.

Powers of the augmentation ideal provide a filtration of the group algebra. Let A(G) := ⨁_{m≥0} I^{m}/I^{m+1} be the associated graded space corresponding to this filtration.

Definition 2.12. An expansion for the group G is a map Z: G → A(G), such that the linear extension Z: 𝔽G → A(G) is filtration preserving and the induced map

gr Z: (gr 𝔽G = A(G)) → (gr A(G) = A(G))

is the identity. An equivalent way to phrase this is that the degree m piece of Z restricted to I^{m} is the projection onto I^{m}/I^{m+1}.

Exercise 2.13. Verify that for the groups PvB_{n} and PwB_{n} the m-th power of the augmentation ideal coincides with the span of all resolutions of m-singular v- or w-braids (by a resolution we mean the formal linear combination where each semivirtual crossing is replaced by the appropriate difference of a virtual and a regular crossing). Then check that the notion of expansion defined above is the same as that of Definition 2.11, restricted to pure braids.

Finally, note the functorial nature of the construction above. What we have described is a functor, called “projectivization” proj : Groups → GradedAlgebras, which assigns to each group G the graded algebra A(G). To each homomorphism ϕ: G → H, proj assigns the induced map gr ϕ: (A(G) = gr 𝔽G) → (A(H) = gr 𝔽H).

2.4. Expansions for w-Braids. The space A_{n}^{w} of arrow diagrams on n strands is an associative algebra in an obvious manner: If the permutations underlying two arrow diagrams are the identity permutations, we simply juxtapose the diagrams. Otherwise we “slide” arrows through permutations in the obvious manner — if τ is a permutation, we declare that τa_{(τi)(τj)} = a_{ij}τ. Instead of seeking an expansion wB_{n} → A_{n}^{w}, we set the bar a little higher and seek a “homomorphic expansion”:

Definition 2.14. A homomorphic expansion Z: wB_{n} → A_{n}^{w} is an expansion that carries products in wB_{n} to products in A_{n}^{w}.

To find a homomorphic expansion, we just need to define it on the generators of wB_{n} and verify that it satisfies the relations defining wB_{n} and the universality condition. Following [BP, Section 5.3] and [AT, Section 8.1] we set Z(�) = � (that is, a transposition in wB_{n} gets mapped to the same transposition in A_{n}^{w}, adding no arrows) and Z(�) = exp(�)�. This last formula is important so deserves to be magnified, explained and replaced by some new notation:

Z (� )= exp (� )·� =  ^{+}  ^{+ \frac{1}{2}}  ^{+ \frac{1}{3!}}  +… =: _{e^{a}} . (15)

Thus the new notation ⟶^{e^{a}} stands for an “exponential reservoir” of parallel arrows, much like e^{a} = 1 + a + aa/2 + aaa/3! + … is a “reservoir” of a’s. With the obvious interpretation for ⟶^{e^{−a}} (the − sign indicates that the terms should have alternating signs, as in e^{−a} = 1 − a + a^{2}/2 − a^{3}/3! + …), the second Reidemeister move �� = 1 forces that we set

Z (� ) = � · exp (−� ) = ^{e^{−a}} = _{e^{−a}} .

Theorem 2.15. The above formulae define an invariant Z: wB_{n} → A_{n}^{w} (that is, Z satisfies all the defining relations of wB_{n}). The resulting Z is a homomorphic expansion (that is, it satisfies the universality property of Definition 2.14).

Proof. (Following [BP, AT]) For the invariance of Z, the only interesting relations to check are the Reidemeister 3 relation of (4) and the Overcrossings Commute relation of (10). For Reidemeister 3, we have

=^{Z} ^{e^{a}}  ^{e^{a}}^{e^{a}} ^{e^{a}}^{e^{a}} ^{e^{a}}  = e^{a_{12}}e^{a_{13}}e^{a_{23}}τ =^{1} e^{a_{12}+a_{13}}e^{a_{23}}τ =^{2} e^{a_{12}+a_{13}+a_{23}}τ,

where τ is the permutation 321 and equality 1 holds because [a_{12}, a_{13}] = 0 by a Tails Commute (TC) relation and equality 2 holds because [a_{12} + a_{13}, a_{23}] = 0 by a \overrightarrow{4T} relation. Likewise, again using TC and \overrightarrow{4T},

=^{Z}  ^{e^{a}}^{e^{a}} ^{e^{a}}  ^{e^{a}}^{e^{a}}^{e^{a}}  = e^{a_{23}}e^{a_{13}}e^{a_{12}}τ = e^{a_{23}}e^{a_{13}+a_{12}}τ = e^{a_{23}+a_{13}+a_{12}}τ,

and so Reidemeister 3 holds. An even simpler proof using just the Tails Commute relation shows that the Overcrossings Commute relation also holds. Finally, since Z is homomorphic, it is enough to check the universality property at degree 1, where it is very easy:

Z (� ) = exp (� ) · � − � = � · � + (terms of degree > 1),

and a similar computation manages the � case. □

Remark 2.16. Note that the main ingredient of the above proof was to show that R := Z(σ_{12}) = e^{a_{12}} satisfies the famed Yang-Baxter equation,

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

where R^{ij} means “place R on strands i and j”.

2.5. Some Further Comments.

2.5.1. Compatibility with Braid Operations. As with any new gadget, we would like to know how compatible the expansion Z of the previous section is with the gadgets we already have; namely, with various operations that are available on w-braids and with the action of w-braids on the free group F_{n} (Section 2.2.3).

[figure: wB_{n}; θ; Z; wB_{n}; Z; A_{n}^{w}; θ; A_{n}^{w}; ⟲]

2.5.1.1. Z is Compatible with Braid Inversion. Let θ denote both the “braid inversion” operation θ: wB_{n} → wB_{n} defined by B ↦ B^{−1} and the “antipode” anti-automorphism θ: A_{n}^{w} → A_{n}^{w} defined by mapping permutations to their inverses and arrows to their negatives (that is, a_{ij} ↦ −a_{ij}). Then the diagram on the right commutes.

[figure: wB_{n}; Δ; Z; wB_{n} × wB_{n}; Z×Z; A_{n}^{w}; Δ; A_{n}^{w} ⊗ A_{n}^{w}; ⟲]

2.5.1.2. Braid Cloning and the Group-Like Property. Let Δ denote both the “braid cloning” operation Δ: wB_{n} → wB_{n} × wB_{n} defined by B ↦ (B, B) and the “co-product” algebra morphism Δ: A_{n}^{w} → A_{n}^{w} ⊗ A_{n}^{w} defined by cloning permutations (that is, τ ↦ τ ⊗ τ) and by treating arrows as primitives (that is, a_{ij} ↦ a_{ij} ⊗ 1 + 1 ⊗ a_{ij}). Then the diagram on the right commutes. In formulae, this is Δ(Z(B)) = Z(B) ⊗ Z(B), which is the statement “Z(B) is group-like”.

[figure: wB_{n}; ι; Z; wB_{n+1}; Z; A_{n}^{w}; ι; A_{n+1}^{w}; ⟲]

2.5.1.3. Strand Insertions. Let ι: wB_{n} → wB_{n+1} be an operation of “inert strand insertion”. Given B ∈ wB_{n}, the resulting ιB ∈ wB_{n+1} will be B with one strand S added at some location chosen in advance — to the left of all existing strands, or to the right, or starting from between the 3rd and the 4th strand of B and ending between the 6th and the 7th strand of B; when adding S, add it “inert”, so that all crossings on it are virtual (this is well defined). There is a corresponding inert strand addition operation ι: A_{n}^{w} → A_{n+1}^{w}, obtained by adding a strand at the same location as for the original ι and adding no arrows. It is easy to check that Z is compatible with ι; namely, that the diagram on the right is commutative.

[figure: wB_{n}; d_{k}; Z; wB_{n−1}; Z; A_{n}^{w}; d_{k}; A_{n−1}^{w}; ⟲]

2.5.1.4. Strand Deletions. Given k between 1 and n, let d_{k}: wB_{n} → wB_{n−1} the operation of “removing the kth strand”. This operation induces a homonymous operation d_{k}: A_{n}^{w} → A_{n−1}^{w}: if D ∈ A_{n}^{w} is an arrow diagram, d_{k}D is D with its kth strand removed if no arrows in D start or end on the kth strand, and it is 0 otherwise. It is easy to check that Z is compatible with d_{k}; namely, that the diagram on the right is commutative.^{14}

14Section ??, “d_{k}: wB_{n} → wB_{n−1}” is an algebraic structure made of two spaces (wB_{n} and wB_{n−1}), two binary operations (braid composition in wB_{n} and in wB_{n−1}), and one unary operation, d_{k}. After projectivization we get the algebraic structure d_{k}: A_{n}^{w} → A_{n−1}^{w} with d_{k} as described above, and an alternative way of stating our assertion is to say that Z is a morphism of algebraic structures. A similar remark applies (sometimes in the negative form) to the other operations discussed in this section.

[figure: F_{n}; �; Z; wB_{n}; Z; _{FA_{n}}; �; A_{n}^{w}; ⟲]

2.5.1.5. Compatibility with the action on F_{n}. Let FA_{n} denote the (degreecompleted) free associative (but not commutative) algebra on generators x_{1}, …, x_{n}. Then there is an “expansion” Z: F_{n} → FA_{n} defined by ξ_{i} ↦ e^{x_{i}} (see [Lin] and the related “Magnus Expansion” of [MKS]). Also, there is a right action of A_{n}^{w} on FA_{n} defined on generators by x_{i}τ = x_{τi} for τ ∈ S_{n} and by x_{j}a_{ij} = [x_{i}, x_{j}] and x_{k}a_{ij} = 0 for k ≠ j and extended by the Leibniz rule to the rest of FA_{n} and then multiplicatively to the rest of A_{n}^{w}.

Exercise 2.17. Using the language of Section ??, verify that FA_{n} = proj F_{n} and that when the actions involved are regarded as instances of the algebraic structure “one monoid acting on another”, we have that (FA_{n}�A_{n}^{w}) = proj (F_{n}�wB_{n}) . Finally, use the definition of the action in (14) and the commutative diagrams of paragraphs 2.5.1.1, 2.5.1.2 and 2.5.1.3 to show that the diagram of paragraph 2.5.1.5 is also commutative.

[figure: wB_{n}; u_{k}; Z; wB_{n+1}; Z; A_{n}^{w}; u_{k}; A_{n+1}^{w}; ⟲̸]^{k  + ^{+} k :=: x + y k  u_{k} u_{k} u_{k}}

2.5.1.6. Unzipping a Strand. Given k between 1 and n, let u_{k}: wB_{n} → wB_{n+1} the operation of “unzipping the kth strand”, briefly defined on the right^{15}. The induced operation u_{k}: A_{n}^{w} → A_{n+1}^{w} is also shown on the right — if an arrow starts (or ends) on the strand being doubled, it is replaced by a sum of two arrows that start (or end) on either of the two “daughter strands” (and this is performed for each arrow independently; so if there are t arrows touching the kth strands in a diagram D, then u_{k}D will be a sum of 2^{t} diagrams).

In some sense, this whole paper as well as the work of Kashiwara and Vergne [KV] and Alekseev and Torossian [AT] is about coming to grips with the fact that Z is not compatible with u_{k} (that the diagram on the right is not commutative). Indeed, let x := a_{13} and y := a_{23} be as on the right, and let s be the permutation 21 and τ the permutation

231. Then d_{1}Z(�) = d_{1}(e^{a_{12}}s) = e^{x+y}τ while Z(d_{1}�) = e^{y}e^{x}τ. So the failure of d_{1} and Z to commute is the ill-behaviour of the exponential function when its arguments are not commuting, which is measured by the BCH formula, central to both [KV] and [AT].

2.5.2. Power and Injectivity. The following theorem is due to Berceanu and Papadima [BP, Theorem 5.4]; a variant of this theorem are also true for ordinary braids [BN2, Ko, HM], and can be proven by similar means:

Theorem 2.18. Z: wB_{n} → A_{n}^{w} is injective. In other words, finite type invariants separate w-braids.

Proof. Follows immediately from the faithfulness of the action F_{n}�wB_{n}, from the compatibility of Z with this action, and from the injectivity of Z: F_{n} → FA_{n} (the latter is well known, see e.g. [MKS, Lin]). Indeed if B_{1} and B_{2} are w-braids and Z(B_{1}) = Z(B_{2}), then Z(ξ)Z(B_{1}) = Z(ξ)Z(B_{2}) for any ξ ∈ F_{n}, therefore ∀ξ Z(ξ ⫽ B_{1}) = Z(ξ ⫽ B_{2}), therefore ∀ξ ξ ⫽ B_{1} = ξ ⫽ B_{2}, therefore B_{1} = B_{2}.

15Unzipping a knotted zipper turns a single band into two parallel ones. This operation is also known as “strand doubling”, but for compatibility with operations that will be introduced later, we prefer “unzipping”.

Remark 2.19. Apart from the obvious, that A_{n}^{w} can be computed degree by degree in exponential time, we do not know a simple formula for the dimension of the degree m piece of A_{n}^{w} or a natural basis of that space. This compares unfavourably with the situation for ordinary braids (see e.g. [BN5]). Also compare with Problem 2.9 and with Remark 2.10.

2.5.3. Uniqueness. There is certainly not a unique expansion for w-braids — if Z_{1} is an expansion and and P is any degree-increasing linear map A^{w} → A^{w} (a “pollution” map), then Z_{2} := (I + P) ∘ Z_{1} is also an expansion, where I: A^{w} → A^{w} is the identity. But that’s all, and if we require a bit more, even that freedom disappears.

Proposition 2.20. If Z_{1,2} : wB_{n} → A_{n}^{w} are expansions then there exists a degree-increasing linear map P: A^{w} → A^{w} so that Z_{2} := (I + P) ∘ Z_{1}.

Proof. (Sketch). Let ŵB_{n} be the unipotent completion of wB_{n}. That is, let ℚwB_{n} be the algebra of formal linear combinations of w-braids, let I be the ideal in ℚwB_{n} be the ideal generated by � = � − � and by � = � − �, and set

ŵB_{n} := \underleftarrow{lim}_{m→∞}ℚwB_{n} /I^{m} .

ŵB_{n} is filtered with F_{m}ŵB_{n} := \underleftarrow{lim}_{m′>m}I^{m} /I^{m′} . An “expansion” can be re-interpreted as

an “isomorphism of ŵB_{n} and A_{n}^{w} as filtered vector spaces”. Always, any two isomorphisms differ by an automorphism of the target space, and that’s the essence of I + P. □

Proposition 2.21. If Z_{1,2} : wB_{n} → A_{n}^{w} are homomorphic expansions that commute with braid cloning (paragraph 2.5.1.2) and with strand insertion (paragraph 2.5.1.3), then Z_{1} = Z_{2}.

Proof. (Sketch). A homomorphic expansion that commutes with strand insertions is determined by its values on the generators �, � and � of wB_{2}. Commutativity with braid cloning implies that these values must be (up to permuting the strands) group like, that is, the exponentials of primitives. But the only primitives are a_{12} and a_{21}, and one may easily verify that there is only one way to arrange these so that Z will respect �^{2} = � · � = 1 and � ↦ � + (higher degree terms). □

2.5.4. The group of non-horizontal flying rings. Let Y_{n} denote the space of all placements of n numbered disjoint oriented unlinked geometric circles in ℝ^{3}. Such a placement is determined by the centres in ℝ^{3} of the circles, the radii, and a unit normal vector for each circle pointing in the positive direction, so dim Y_{n} = 3n + n + 3n = 7n. S_{n} ⋉ ℤ_{2}^{n} acts on Y_{n} by permuting the circles and mapping each circle to its image in either an orientation-preserving or an orientation-reversing way. Let Ỹ_{n} denote the quotient Y_{n}/S_{n} ⋉ ℤ_{2}^{n}. The fundamental group π_{1}(Ỹ_{n}) can be thought of as the “group of flippable flying rings”. Without loss of generality, we can assume that the basepoint is chosen to be a horizontal placement. We want to study the relationship of this group to wB_{n}.

Theorem 2.22. π_{1}(Ỹ_{n}) is a ℤ_{2}^{n}-extension of wB_{n}, generated by s_{i}, σ_{i} (1 ≤ i ≤ n − 1), and w_{i} (“flips”), for 1 ≤ i ≤ n; with the relations as above, and in addition:

w_{i}^{2} = 1;  w_{i}w_{j} = w_{j}w_{i};  w_{j}s_{i} = s_{i}w_{j} when i ≠ j, j + 1;
w_{i}s_{i} = s_{i}w_{i+1}; w_{i+1}s_{i} = s_{i}w_{i};

w_{j}σ_{i} = σ_{i}w_{j} if j ≠ i, i + 1; w_{i+1}σ_{i} = σ_{i}w_{i}; yet w_{i}σ_{i} = s_{i}σ_{i}^{−1}s_{i}w_{i+1}.

The two most interesting flip relations in pictures:

=^{=}^{w}_{i}  ^{w}_{i}  _{i}_{w}  ^{w}_{i}_{i+1}  _{i+1}  _{i+1}  _{i+1}  (16)

w_{i} = w ^{i}

Instead of a proof, we provide some heuristics. Since each circle starts out in a horizontal position and returns to a horizontal position, there is an integer number of “flips” they do in between, these are the generators w_{i}, as shown on the right.

The first relation says that a double flip is homotopic to doing nothing. Technically, there are two different directions of flips, and they are the same via this (non-obvious but true) relation. The rest of the first line is obvious: flips of different rings commute, and if two rings fly around each other while another one flips, the order of these events can be switched by homotopy. The second line says that if two rings trade places with no interaction while one flips, the order of these events can be switched as well. However, we have to re-number the flip to conform to the strand labelling convention.

The only subtle point is how flips interact with crossings. First of all, if one ring flies through another while a third one flips, the order clearly does not matter. If a ring flies through another and also flips, the order can be switched. However, if ring A flips and then ring B flies through it, this is homotopic to ring B flying through ring A from the other direction and then ring A flipping. In other words, commuting σ_{i} with w_{i} changes the “sign of the crossing” in the sense of Exercise 2.7. This gives the last, and the only truly non-commutative flip relation.

To explain why the flip is denoted by w, let us consider the alternative description by ribbon tubes. A flipping ring traces a so called wen^{16} in ℝ^{4}. A wen is a Klein bottle cut along a meridian circle, as shown. The wen is embedded in ℝ^{4}.

Finally, note that π_{1}Y_{n} is exactly the pure w-braid group PwB_{n}: since each ring has to return to its original position and orientation, each does an even number of flips. The flips (or wens) can all be moved to the bottoms of the braid diagram strands (to the bottoms of the tubes, to the beginning of words), at a possible cost, as specified by Equation (16). Once together, they pairwise cancel each other. As a result, this group can be thought of as not containing wens at all.

2.5.5. The Relationship with u-Braids. For the sake of ignoring strand permutations, we restrict our attention to pure braids.

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

By Section 2.3.2, for any expansion Z^{u}: PuB_{n} → A_{n}^{u} (where PuB_{n} is the “usual” braid group and A_{n}^{u} is the algebra of horizontal chord diagrams on n strands), there is a square of maps as shown on the right. Here Z^{w} is the expansion constructed in Section 2.4, the left vertical map a is the composition of the inclusion and projection maps PuB_{n} → PvB_{n} → PwB_{n}. The map α is the induced map by the functoriality of projectivization, as noted after Exercise 2.13. The reader can verify that α maps each chord to the sum of its two possible directed versions.

16The term wen was coined by Kanenobu and Shima in [KS]

Note that this square is not commutative for any choice of Z^{u} even in degree 2: the image of a crossing under Z^{w} is outside the image of α.

P uB_{n} P wB_{n}  A_{n}^{u} A_{n}^{w} Z^{w}  α a  Z_{c}^{u}

More specifically, for any choice c of a “parenthesization” of n points, the KZ-construction / Kontsevich integral (see for example [BN3]) returns an expansion Z_{c}^{u} of u-braids. As we shall see in Proposition ??, for any choice of c, the two compositions α ∘ Z_{c}^{u} and Z^{w} ∘ a are “conjugate in a bigger space”: there is a map i from A^{w} to a larger space of “non-horizontal arrow diagrams”, and in this space the images of the above composites are conjugate. However, we are not certain that i is an injection, and whether the conjugation leaves the i-image of A^{w} invariant, and so we do not know if the two compositions differ merely by an outer automorphism of A^{w}.

3. w-Knots

Section Summary. In 3.1 we define v-knots and w-knots (long v-knots and long w-knots, to be precise) and discuss the map v → w. In 3.2 we determine the space of “chord diagrams” for w-knots to be the space A^{w}(↑) of arrow diagrams modulo \overrightarrow{4T} and TC relations and in 3.3 we compute some relevant dimensions. In 3.5 we show that A^{w}(↑) can be re-interpreted as a space of trivalent graphs modulo STU- and IHX-like relations, and is therefore related to Lie algebras (Sec. 3.6). This allows us to completely determine A^{w}(↑). With no difficulty at all in 3.4 we construct a universal finite type invariant for w-knots. With a bit of further difficulty we show in Sec. 3.7 that it is essentially equal to the Alexander polynomial.

Knots are the wrong objects for study in knot theory, v-knots are the wrong objects for study in the theory of v-knotted objects and w-knots are the wrong objects for study in the theory of w-knotted objects. Studying uvw-knots on their own is the parallel of studying cakes and pastries as they come out of the bakery — we sure want to make them our own, but the theory of desserts is more about the ingredients and how they are put together than about the end products. In algebraic knot theory this reflects through the fact that knots are not finitely generated in any sense (hence they must be made of some more basic ingredients), and through the fact that there are very few operations defined on knots (connected sums and satellite operations being the main exceptions), and thus most interesting properties of knots are transcendental, or non-algebraic, when viewed from within the algebra of knots and operations on knots [BN8].

The right objects for study in knot theory, or v-knot theory or w-knot theory, are thus the ingredients that make up knots and that permit a richer algebraic structure. These are braids, studied in the previous section, and even more so tangles and tangled graphs, studied in the following sections. Yet tradition has its place and the sweets are tempting, and we feel compelled to introduce some of the tools we will use in the deeper and healthier study of w-tangles and w-tangled foams in the limited but tasty arena of the baked goods of knot theory, the knots themselves.

3.1. v-Knots and w-Knots. v-Knots may be understood either as knots drawn on surfaces modulo the addition or removal of empty handles [Ka2, Kup] or as “Gauss diagrams” (Remark 3.4), or simply “unembedded but wired together” crossings modulo the Reidemeister moves ([Ka2, Rou] and Section ??). But right now we forgo the topological and the abstract and give only the “planar” (and somewhat less philosophically satisfying) definition of v-knots.

Figure 5. A long v-knot diagram with 2 virtual crossings, 2 positive crossings and 2 negative crossings. A positive-negative pair can easily be cancelled using R2, and then a virtual crossing can be cancelled using VR1, and it seems that the rest cannot be simplified any further.

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

Figure 6. The relations defining v-knots and w-knots, along with two relations that are not imposed.

Definition 3.1. A “long v-knot diagram” is an arc smoothly drawn in the plane from −∞ to +∞, with finitely many self-intersections, divided into “virtual crossings” � and over- and under-crossings, � and �, and regarded up to planar isotopy. A picture is worth more than a more formal definition, and one appears in Figure 5. A “long v-knot” is an equivalence class of long v-knot diagrams, modulo the equivalence generated by the Reidemeister 1^{s}, 2 and 3 moves (R1^{s}, R2 and R3)^{17}, the virtual Reidemeister 1 through 3 moves (VR1, VR2, VR3), and by the mixed relations (M); all these are shown in Figure 6. Finally, “long w-knots” are obtained from long v-knots by also dividing by the Overcrossings Commute (OC) relations, also shown in Figure 6. Note that we never mod out by the Reidemeister 1 (R1) move nor by the Undercrossings Commute relation (UC).

Definition and Warning 3.2. A “circular v-knot” is like a long v-knot, except parametrized by a circle rather than by a long line. Unlike the case of ordinary knots, circular v-knots are not equivalent to long v-knots. The same applies to w-knots.

Definition and Warning 3.3. Long v-knots form a monoid using the concatenation operation #. Unlike the case of ordinary knots, the resulting monoid is not Abelian. The same applies to w-knots.

Remark 3.4. A “Gauss diagram” is a straight “skeleton line” along with signed directed chords (signed “arrows”) marked along it (more at [Ka2, GPV]). Gauss diagrams are in an obvious bijection with long v-knot diagrams; the skeleton line of a Gauss diagram corresponds to the parameter space of the v-knot, and the arrows correspond to the crossings, with each arrow heading from the upper strand to the lower strand, marked by the sign of the relevant crossing:

− +  +−^{2} ^{4} ^{3}^{1}

One may also describe the relations in Figure 6 as well as circular v-knots and other types of v-knots (as we will encounter later) in terms of Gauss diagrams with varying skeletons.

17 R1^{s} is the “spun” version of R1 — kinks can be spun around, but not removed outright. See Figure 6.

Figure 7. The positive and negative under-then-over kinks (left), and the positive and negative over-then-under kinks (right). In each pair the negative kink is the #-inverse of the positive kink.

Remark 3.5. Since we do not mod out by R1, it is perhaps more appropriate to call our class of v/w-knots “framed long v/w-knots”, but since we care more about framed v/w-knots than about unframed ones, we reserve the unqualified name for the framed case, and when we do wish to mod out by R1 we will explicitly write “unframed long v/w-knots”.

Recall that in the case of “usual knots”, or u-knots, dropping the R1 relation altogether also results in a ℤ^{2}-extension of unframed knot theory, where the two factors of ℤ are framing and rotation number. If one wants to talk about “true” framed knots, one mods out by the spun Reidemeister 1 relation (R1^{s} of Figure 6), which preserves the blackboard framing but does not preserve the rotation number. We take the analogous approach here, including the R1^{s} relation but not R1 also in the v and w cases.

This said, note that the monoid of long v-knots is just a central extension by ℤ of the monoid of unframed long v-knots, and so studying the framed case is not very different from studying the unframed case. Indeed the four “kinks” of Figure 7 generate a central ℤ within long v-knots, and it is not hard to show that the sequence

1 ⟶ ℤ ⟶ {long v-knots} ⟶ {unframed long v-knots} ⟶ 1  (17)

is split and exact. The same can be said for w-knots.

Exercise 3.6. Show that a splitting of the sequence (17) is given by the “self-linking” invariant sl: {long v-knots} → ℤ defined by

sl(K) := ∑_{crossings ; x in K} sign x,

where K is a v-knot diagram, and the sign of a crossing x is defined so as to agree with the signs in Figure 7.

Remark 3.7. w-Knots are strictly weaker than v-knots — a notorious example is the Kishino knot (e.g. [Dye]) which is non-trivial as a v-knot yet both it and its mirror are trivial as w-knots. Yet ordinary knots inject even into w-knots, as the Wirtinger presentation makes sense for w-knots and therefore w-knots have a “fundamental quandle” which generalizes the fundamental quandle of ordinary knots [Ka2], and as the fundamental quandle of ordinary knots separates ordinary knots [Joy].

3.1.1. A topological construction of Satoh’s tubing map. Following Satoh [Sa] and using the same constructions as in Section 2.2.2, we can map w-knots to (“long”) ribbon tubes in ℝ^{4} (and the relations in Figure 6 still hold). It is natural to expect that this “tubing” map is an isomorphism; in other words, that the theory of w-knots provides a “Reidemeister framework” for long ribbon tubes in ℝ^{4} — that every long ribbon tube is in the image of this map and that two “w-knot diagrams” represent the same long ribbon tube iff they differ by a sequence of moves as in Figure 6. This remains unproven.

Let δ: {v-knots} → {Ribbon tori in ℝ^{4}} denote the tubing map described in Section 2.2.2. In Satoh’s [Sa] δ is called “Tube”. It is worthwhile to give a completely “topological” definition of δ. To do this we must start with a topological interpretation of v-knots.

The standard topological interpretation of v-knots (e.g. [Kup]) is that they are oriented framed knots drawn^{18} on an oriented surface Σ, modulo “stabilization”, which is the addition and/or removal of empty handles (handles that do not intersect with the knot). We prefer an equivalent, yet even more bare-bones approach. For us, a virtual knot is an oriented framed knot γ drawn on a “virtual surface Σ for γ”. More precisely, Σ is an oriented surface that may have a boundary, γ is drawn on Σ, and the pair (Σ, γ) is taken modulo the following relations:

• Isotopies of γ on Σ (meaning, in Σ × [−ϵ, ϵ]).

• Tearing and puncturing parts of Σ away from γ:

tearing  ∂Σ Σ γ  γ  isotopy  puncturing

(We call Σ a “virtual surface” because tearing and puncturing imply that we only care about it in the immediate vicinity of γ).

We can now define^{19} a map δ, defined on v-knots and taking values in ribbon tori in ℝ^{4}: given (Σ, γ), embed Σ arbitrarily in ℝ_{xzt}^{3} ⊂ ℝ^{4}. Note that the unit normal bundle of Σ in ℝ^{4} is a trivial circle bundle and it has a distinguished trivialization, constructed using its positive-y-direction section and the orientation that gives each fibre a linking number +1 with the base Σ. We say that a normal vector to Σ in ℝ^{4} is “near unit” if its norm is between 1 − ϵ and 1 + ϵ. The near-unit normal bundle of Σ has as fibre an annulus that can be identified with [−ϵ, ϵ] × S^{1} (identifying the radial direction [1 − ϵ, 1 + ϵ] with [−ϵ, ϵ] in an orientation-preserving manner), and hence the near-unit normal bundle of Σ defines an embedding of Σ × [−ϵ, ϵ] × S^{1} into ℝ^{4}. On the other hand, γ is embedded in Σ × [−ϵ, ϵ] so γ × S^{1} is embedded in Σ × [−ϵ, ϵ] × S^{1}, and we can let δ(γ) be the composition

γ × S^{1} ↪ Σ × [−ϵ, ϵ] × S^{1} ↪ ℝ^{4},

which is a torus in ℝ^{4}, oriented using the given orientation of γ and the standard orientation of S^{1}.

A framing of a knot (or a v-knot) γ can be thought of as a “nearby companion” to γ. Applying the above procedure to a knot and a nearby companion simultaneously, we find that δ takes framed v-knots to framed ribbon tori in ℝ^{4}, where a framing of a tube in ℝ^{4} is a continuous up-to-homotopy choice of unit normal vector at every point of the tube. Note that from the perspective of flying rings as in Section 2.2.1 a framing is a “companion ring” to a flying ring. In the framing of δ(γ) the companion ring is never linked with the main ring, but can fly parallel inside, outside, above or below it and change these positions, as shown below.

18Here and below, “drawn on Σ” means “embedded in Σ × [−ϵ, ϵ]”.

19Following a private discussion with Dylan Thurston.

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

Figure 8. An arrow diagram of degree 6, a 6T relation, and an RI relation.

We leave it to the reader to verify that δ(γ) is ribbon, that it is independent of the choices made within its construction, that it is invariant under isotopies of γ and under tearing and puncturing of Σ, that it is also invariant under the “overcrossings commute” relation of Figure 6 and hence the true domain of δ is w-knots, and that it is equivalent to Satoh’s tubing map.

3.2. Finite Type Invariants of v-Knots and w-Knots. Much as for v-braids and wbraids (Section 2.3) and much as for ordinary knots (e.g. [BN1]) we define finite type invariants for v-knots and for w-knots using an alternation scheme with � → � − � and � → �−�. That is, we extend any Abelian-group-valued invariant of v- or w-knots to v- or w-knots also containing “semi-virtual crossings” like � and � using the above assignments, and we declare an invariant to be “of type m” if it vanishes on v- or w-knots with more than m semi-virtuals. As for v- and w-braids and as for ordinary knots, such invariants have an “mth derivative”, their “weight system”, which is a linear functional on the space A^{sv}(↑) (for v-knots) or A^{sw}(↑) (for w-knots). We turn to the definitions of these spaces, following [GPV, BHLR]:

Definition 3.8. An “arrow diagram” is a chord diagram along a long line (called “the skeleton”), in which the chords are oriented (hence “arrows”). An example is in Figure 8. Let D^{v}(↑) be the space of formal linear combinations of arrow diagrams. Let A^{v}(↑) be D^{v}(↑) modulo all “6T relations”. Here a 6T relation is any (signed) combination of arrow diagrams obtained from the diagrams in Figure 3 by placing the 3 vertical strands there along a long line in any order, and possibly adding some further arrows in between. An example is in Figure 8. Let A^{sv}(↑) be the further quotient of A^{v}(↑) by the RI relation, where the RI (for Rotation number Independence) relation asserts that an isolated arrow pointing to the right equals an isolated arrow pointing to the left, as shown in Figure 8^{20}.

20 The XII relation of [BHLR] follows from RI and need not be imposed.

and + =^{\overrightarrow{4T}}  =^{TC} +
Figure 9. The TC and the \overrightarrow{4T} relations for knots.

Let A^{w}(↑) be the further quotient of A^{v}(↑) by the “Tails Commute” (TC) relation, first displayed in Figure 4 and reproduced for the case of a long-line skeleton in Figure 9. Likewise, let A^{sw}(↑) := A^{sv}(↑)/T C = A^{w}(↑)/RI. Alternatively, noting that given TC two of the terms in 6T drop out, A^{w}(↑) is the space of formal linear combinations of arrow diagrams modulo TC and \overrightarrow{4T} relations, displayed in Figures 4 and 9. Likewise, A^{sw} = D^{v}/T C, \overrightarrow{4T}, RI. Finally, grade D^{v}(↑) and all of its quotients by declaring that the degree of an arrow diagram is the number of arrows in it.

As an example, the spaces A^{v,sv,w,sw}(↑) restricted to degrees up to 2 are studied in detail in Section 4.2.

In the same manner as in the theory of finite type invariants of ordinary knots (see especially [BN1, Section 3], the spaces A^{−}(↑) carry much algebraic structure. The obvious juxtaposition product makes them into graded algebras. The product of two finite type invariants is a finite type invariant (whose type is the sum of the types of the factors); this induces a product on weight systems, and therefore a co-product Δ on arrow diagrams. In brief (and much the same as in the usual finite type story), the co-product ΔD of an arrow diagram D is the sum of all ways of dividing the arrows in D between a “left co-factor” and a “right co-factor”. In summary,

Proposition 3.9. A^{v}(↑), A^{sv}(↑), A^{w}(↑), and A^{sw}(↑) are co-commutative graded bi-algebras.

By the Milnor-Moore theorem [MM] we find that A^{v,sv,w,sw}(↑) are the universal enveloping algebras of their Lie algebras of primitive elements. Denote these (graded) Lie algebras by P^{v,sv,w,sw}(↑), respectively.

When we grow up we’d like to understand A^{v}(↑) and A^{sv}(↑). At the moment we know only very little about these spaces beyond the generalities of Proposition 3.9. In the next section some dimensions of low degree parts of A^{v,sv}(↑) are displayed. Also, given a finite dimensional Lie bialgebra and a finite dimensional representation thereof, we know how to construct linear functionals on A^{v}(↑) (one in each degree) [Hav, Leu] (but not on A^{sv}(↑)). But we don’t even know which degree m linear functionals on A^{sv}(↑) are the weight systems of degree m invariants of v-knots (that is, we have not solved the “Fundamental Problem” [BS] for v-knots).

As we shall see below, the situation is much brighter for A^{w,sw}(↑).

3.3. Some Dimensions. The table below lists what we could find about A^{v} and A^{w} by crude brute force computations in low degrees. We list degrees 0 through 7. The spaces we study are A^{−}(↑), A^{s−}(↑), A^{r−}(↑) which is A^{−}(↑) moded out by “isolated” arrows ^{21}, P^{−}(↑) which is the space of primitives in A^{−}(↑), and A^{−}(◯), A^{s−}(◯), and A^{r−}(◯), which are the same as A^{−}(↑), A^{s−}(↑), and A^{r−}(↑) except with closed knots (knots with a circle skeleton) replacing long knots. Each of these spaces we study in three variants: the “v” and the “w” variants, as well as the \underline{u}sual knots “u” variant which is here just for comparison. We also include a row “dim G_{m}Lie^{−}(↑)” for the dimensions of “Lie-algebraic weight systems”. Those are explained in the u and v cases in [BN1, Hav, Leu], and in the w case in Section 3.6.

See Section 4.2

m  0  1  2  3  4  5  6  7 Comments
dim G_{m}A^{−}(↑) \begin{matrix}u | v \\ w \end{matrix} \begin{matrix}1 | 1 \\ 1 \end{matrix} \begin{matrix}1 | 2 \\ 2 \end{matrix} \begin{matrix}2 | 7 \\ 4 \end{matrix} \begin{matrix}3 | 27 \\ 7 \end{matrix} \begin{matrix}6 | 139 \\ 12 \end{matrix} \begin{matrix}10 | 813 \\ 19 \end{matrix} \begin{matrix}19 |? \\ 30 \end{matrix} \begin{matrix}33 |? \\ 45 \end{matrix} \begin{matrix}1 | 2 \\ 3, 4, 5 \end{matrix}
dim G_{m}Lie^{−}(↑) \begin{matrix}u | v \\ w \end{matrix} \begin{matrix}1 | 1 \\ 1 \end{matrix} \begin{matrix}1 | 2 \\ 2 \end{matrix} \begin{matrix}2 | 7 \\ 4 \end{matrix} \begin{matrix}3 | 27 \\ 7 \end{matrix} \begin{matrix}6 | ≥128 \\ 12 \end{matrix} \begin{matrix}10 |? \\ 19 \end{matrix} \begin{matrix}19 |? \\ 30 \end{matrix} \begin{matrix}33 |? \\ 45 \end{matrix}  \begin{matrix}1 | 6 \\ 5 \end{matrix}
dim G_{m}A^{s−}(↑) \begin{matrix}u | v \\ w \end{matrix} \begin{matrix}− | 1 \\ 1 \end{matrix} \begin{matrix}− | 1 \\ 1 \end{matrix} \begin{matrix}− | 3 \\ 2 \end{matrix} \begin{matrix}− | 10 \\ 3 \end{matrix} \begin{matrix}− | 52 \\ 5 \end{matrix} \begin{matrix}− | 298 \\ 7 \end{matrix} \begin{matrix}− |? \\ 11 \end{matrix} \begin{matrix}− |? \\ 15 \end{matrix}  \begin{matrix}7 | 2 \\ 3, 8 \end{matrix}
dim G_{m}A^{r−}(↑) \begin{matrix}u | v \\ w \end{matrix} \begin{matrix}1 | 1 \\ 1 \end{matrix} \begin{matrix}0 | 0 \\ 0 \end{matrix} \begin{matrix}1 | 2 \\ 1 \end{matrix} \begin{matrix}1 | 7 \\ 1 \end{matrix} \begin{matrix}3 | 42 \\ 2 \end{matrix} \begin{matrix}4 | 246 \\ 2 \end{matrix} \begin{matrix}9 |? \\ 4 \end{matrix} \begin{matrix}14 |? \\ 4 \end{matrix} \begin{matrix}1 | 9 \\ 3, 10 \end{matrix}
dim G_{m}P^{−}(↑) \begin{matrix}u | v \\ w \end{matrix} \begin{matrix}0 | 0 \\ 0 \end{matrix} \begin{matrix}1 | 2 \\ 2 \end{matrix} \begin{matrix}1 | 4 \\ 1 \end{matrix} \begin{matrix}1 | 15 \\ 1 \end{matrix} \begin{matrix}2 | 82 \\ 1 \end{matrix} \begin{matrix}3 | 502 \\ 1 \end{matrix} \begin{matrix}5 |? \\ 1 \end{matrix} \begin{matrix}8 |? \\ 1 \end{matrix}  \begin{matrix}1 | 11 \\ 3 \end{matrix}
dim G_{m}A^{−}(◯) \begin{matrix}u | v \\ w \end{matrix} \begin{matrix}1 | 1 \\ 1 \end{matrix} \begin{matrix}1 | 1 \\ 1 \end{matrix} \begin{matrix}2 | 2 \\ 1 \end{matrix} \begin{matrix}3 | 5 \\ 1 \end{matrix} \begin{matrix}6 | 19 \\ 1 \end{matrix} \begin{matrix}10 | 77 \\ 1 \end{matrix} \begin{matrix}19 |? \\ 1 \end{matrix} \begin{matrix}33 |? \\ 1 \end{matrix} \begin{matrix}1 | 12 \\ 3 \end{matrix}
dim G_{m}A^{s−}(◯) \begin{matrix}u | v \\ w \end{matrix} \begin{matrix}− | 1 \\ 1 \end{matrix} \begin{matrix}− | 1 \\ 1 \end{matrix} \begin{matrix}− | 1 \\ 1 \end{matrix} \begin{matrix}− | 2 \\ 1 \end{matrix} \begin{matrix}− | 6 \\ 1 \end{matrix} \begin{matrix}− | 23 \\ 1 \end{matrix} \begin{matrix}− |? \\ 1 \end{matrix} \begin{matrix}− |? \\ 1 \end{matrix}  \begin{matrix}7 | 2 \\ 3 \end{matrix}
dim G_{m}A^{r−}(◯) \begin{matrix}u | v \\ w \end{matrix} \begin{matrix}1 | 1 \\ 1 \end{matrix} \begin{matrix}0 | 0 \\ 0 \end{matrix} \begin{matrix}1 | 0 \\ 0 \end{matrix} \begin{matrix}1 | 1 \\ 0 \end{matrix}  \begin{matrix}3 | 4 \\ 0 \end{matrix}  \begin{matrix}4 | 17 \\ 0 \end{matrix} \begin{matrix}9 |? \\ 0 \end{matrix} \begin{matrix}14 |? \\ 0 \end{matrix} \begin{matrix}1 | 12 \\ 3 \end{matrix}

Comments 3.10. (1) Much more is known computationally on the u-knots case. See

especially [BN1, BN4, Kn, AS].

(2) These dimensions were computed by Louis Leung and DBN using a program available at [BND2, “Dimensions”].

(3) As we shall see in Section 3.5, the spaces associated with w-knots are understood to all degrees.

(4) To degree 4, these numbers were also verified by [BND2, “Dimensions”].

(5) The next few numbers in these sequences are 67, 97, 139, 195, 272.

(6) These dimensions were computed by Louis Leung and DBN using a program available at [BND2, “Arrow Diagrams and gl(N)”]. Note the match with the row above.

(7) There is no “s” quotient in the “u” case.

(8) The next few numbers in this sequence are 22, 30, 42, 56, 77.

(9) These numbers were computed by [BND2, “Dimensions”]. Contrary to the A^{u} case, A^{rv} is not the quotient of A^{v} by the ideal generated by degree 1 elements, and therefore the dimensions of the graded pieces of these two spaces cannot be deduced from each other using the Milnor-Moore theorem.

21That is, A^{r−}(↑) is A^{−}(↑) modulo “Framing Independence” (FI) relations [BN1], with the isolated arrow taken with either orientation. It is the space related to finite type invariants of unframed knots, on which the first Reidemeister move is also imposed, in the same way as A^{−}(↑) is related to framed knots.

(10) The next few numbers in this sequence are 7,8,12,14,21.

(11) These dimensions were deduced from the dimensions of G_{m}A^{v}(↑) using the Milnor-Moore theorem.

(12) Computed by [BND2, “Dimensions”]. Contrary to the A^{u} case, A^{v}(◯), A^{sv}(◯), and A^{rv}(◯) are not isomorphic to A^{v}(↑), A^{sv}(↑), and A^{rv}(↑) and separate computations are required.

3.4. Expansions for w-Knots. The notion of “an expansion” (or “a universal finite type invariant”) for w-knots (or v-knots) is defined in complete analogy with the parallel notion for ordinary knots (e.g. [BN1]), except replacing double points (�) with semi-virtual crossings (� and �) and replacing chord diagrams by arrow diagrams. Alternatively, it is the same as an expansion for w-braids (Definition 2.11), with the obvious replacement of w-braids by wknots. Just as in the cases of ordinary knots and/or w-braids, the existence of an expansion Z: {w-knots} → A^{sw}(↑) is equivalent to the statement “every weight system integrates”, i.e., “every degree m linear functional on A^{sw}(↑) is the mth derivative of a type m invariant of long w-knots”.

Theorem 3.11. There exists an expansion Z: {w-knots} → A^{sw}(↑).

Proof. It is best to define Z by an example, and it is best to display the example only as a picture:

Z  =  = e^{−a}  e^{a} e^{a} e^{−a}  e^{−a} e^{a}  e^{a} e^{−a} e^{a}^{2}^{1}  ^{4} ^{3} e^{−a}

It is clear how to define Z(K) in the general case — for every crossing in K place an exponential reservoir of arrows (compare with (15)) next to that crossing, with the arrows heading from the upper strand to the lower strand, taking positive reservoirs (e^{a}, with a symbolizing the arrow) for positive crossings and negative reservoirs (e^{−a}) for negative crossings, and then tug the skeleton until it looks like a straight line. Note that the Tails Commute relation in A^{sw} is used to show that all reasonable ways of placing an arrow reservoir at a crossing (with its heading and sign fixed) are equivalent:

=  =  =  e^{a} e^{a}  e^{a} e^{a}

The same proof that shows the invariance of Z in the braids case (Theorem 2.15) works here as well^{22}, and the same argument as in the braids case shows the universality of Z. □

Remark 3.12. Using the language of Gauss diagrams (Remark 3.4) the definition of Z is even simpler. Simply map every positive arrow in a Gauss diagram to a positive (e^{a}) reservoir,

22A tiny bit of extra care is required for invariance under R1^{s}: it easily follows from RI.

\begin{aligned}& l \\ r \end{aligned}

Figure 10. A degree 11 w-Jacobi diagram on a long line skeleton. It has a skeleton line at the bottom, 13 vertices along the skeleton (of which 2 are incoming and 11 are outgoing), 9 internal vertices (with only one explicitly marked with “left” (l) and “right” (r)) and one bubble. The five quadrivalent vertices that seem to appear in the diagram are just projection artifacts and graph-theoretically, they don’t exist.

and every negative one to a negative (e^{−a}) reservoir:

Z − +  +−  e^{−a} e^{a}  e^{a} e^{−a}

An expansion (a universal finite type invariant) is as interesting as its target space, for it is just a tool that takes linear functionals on the target space to finite type invariants on its domain space. The purpose of the next section is to find out how interesting are our present target space, A^{sw}(↑), and its “parent”, A^{w}(↑).

3.5. Jacobi Diagrams, Trees and Wheels. In studying A^{w}(↑) we again follow the model set by ordinary knots. Compare the following definitions and theorem with [BN1, Section 3].

Definition 3.13. A “w-Jacobi diagram on a long line skeleton”^{23} is a connected graph made of the following ingredients:

• A “long” oriented “skeleton” line. We usually draw the skeleton line a bit thicker for emphasis.

• Other directed edges, usually called “arrows”.

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

• Trivalent “internal vertices” in which two arrows end and one arrow begins. The internal vertices are “oriented” — of the two arrows that end in an internal vertices, one is marked as “left” and the other is marked as “right”. In reality when a diagram is drawn in the plane, we almost never mark “left” and “right”, but instead assume the “left” and “right” inherited from the plane, as seen from the outgoing arrow from the given vertex.

Note that we allow multiple arrows connecting the same two vertices (though at most two are possible, given connectedness and trivalence) and we allow “bubbles” — arrows that begin and end in the same vertex. Note that for the purpose of determining equality of diagrams the skeleton line is distinguished. The “degree” of a w-Jacobi diagram is half the number of trivalent vertices in it, including both internal and skeleton vertices. An example of a w-Jacobi diagram is in Figure 10.

23What a mouthful! We usually short this to “w-Jacobi diagram”, or sometimes “arrow diagram” or just “diagram”.

=  ^{−}  ^{=}  ^{−}^{\overrightarrow{ST U}_{1}:}  ^{\overrightarrow{ST U}_{2}:}^{e}  ^{e}  ^{e}  ^{e}  ^{e}  ^{e}^{=}  ^{−} 0^{\overrightarrow{ST U}_{3} =TC:}  ^{e}  ^{e}
Figure 11. The \overrightarrow{ST U}_{1,2} and TC relations with their “central edges” marked e.

\overrightarrow{AS}:  0 = ^{l} ^{r} ^{+} ^{r} ^{l}  ^{\overrightarrow{IHX}:}  ^{e}  ^{=}  ^{e}  ^{−}  ^{e}
Figure 12. The \overrightarrow{AS} and \overrightarrow{IHX} relations.

Definition 3.14. Let D^{wt}(↑) be the graded vector space of formal linear combinations of

w-Jacobi diagrams on a long line skeleton, and let A^{wt}(↑) be D^{wt}(↑) modulo the \overrightarrow{ST U}_{1},

\overrightarrow{ST U}_{2}, and TC relations of Figure 11. Note that that each diagram appearing in each \overrightarrow{ST U} relation has a “central edge” e which can serve as an “identifying name” for that \overrightarrow{ST U}. Thus given a diagram D with a marked edge e which is either on the skeleton or which contacts the skeleton, there is an unambiguous \overrightarrow{ST U} relation “around” or “along” the edge e.

We like to call the following theorem “the bracket-rise theorem”, for it justifies the introduction of internal vertices, and as should be clear from the \overrightarrow{ST U} relations and as will become even clearer in Section 3.6, internal vertices can be viewed as “brackets”. Two other bracket-rise theorems are Theorem 6 of [BN1] and Ohtsuki’s theorem, Theorem 4.9 of [Po].

Theorem 3.15 (bracket-rise). The obvious inclusion ι: D^{v}(↑) → D^{wt}(↑) of arrow diagrams (Definition 3.8) into w-Jacobi diagrams descends to the quotient A^{w}(↑) and induces an iso-

morphism ῑ: A^{w}(↑) ⟶^{∼} A^{wt}(↑). Furthermore, the \overrightarrow{AS} and \overrightarrow{IHX} relations of Figure 12 hold in A^{wt}(↑).

Proof. The proof, joint with D. Thurston, is modelled after the proof of Theorem 6 of [BN1]. To show that ι descends to A^{w}(↑) we just need to show that in A^{wt}(↑), \overrightarrow{4T} follows from \overrightarrow{ST U}_{1,2}. Indeed, applying \overrightarrow{ST U}_{1} along the edge e_{1} and \overrightarrow{ST U}_{2} along e_{2} in the picture below, we get the two sides of \overrightarrow{4T}:

=^{=}  ^{−}^{−}^{\overrightarrow{ST U}_{1}}^{\overrightarrow{ST U}_{2}}^{e_{2}}  ^{e_{1}}  (18)

The fact that ῑis surjective is obvious; indeed, for diagrams in A^{wt}(↑) that have no internal vertices there is nothing to show, for they are really in A^{w}(↑). Further, by repeated use of \overrightarrow{ST U}_{1,2} relations, all internal vertices in any diagram in A^{wt}(↑) can be removed (remember that the diagrams in A^{wt}(↑) are always connected, and in particular, if they have an internal vertex they must have an internal vertex connected by an edge to the skeleton, and the latter vertex can be removed first).

To complete the proof that ῑ is an isomorphism it is enough to show that the “elimination of internal vertices” procedure of the last paragraph is well defined — that its output is independent of the order in which \overrightarrow{ST U}_{1,2} relations are applied in order to eliminate internal vertices. Indeed, this done, the elimination map would by definition satisfy the \overrightarrow{ST U}_{1,2} relations and thus descend to a well defined inverse for ῑ.

On diagrams with just one internal vertex, Equation (18) shows that all ways of eliminating that vertex are equivalent modulo \overrightarrow{4T} relations, and hence the elimination map is well defined on such diagrams.

Now assume that we have shown that the elimination map is well defined on all diagrams with at most 7 internal vertices, and let D be a diagram with 8 internal vertices^{24}. Let e and e′ be edges in D that connect the skeleton of D to an internal vertex. We need to show that any elimination process that begins with eliminating e yields the same answer, modulo \overrightarrow{4T}, as any elimination process that begins with eliminating e′. There are several cases to consider.

e  e′

Case I. e and e′ connect the skeleton to different internal vertices of D. In this case, after eliminating e we get a signed sum of two diagrams with exactly 7 internal vertices, and since the elimination process is well defined on such diagrams, we may as well continue by eliminating e′ in each of those, getting a signed sum of 4 diagrams with 6 internal vertices each. On the other hand, if we start by eliminating e′ we can continue by eliminating e, and we get the same signed sum of 4 diagrams with 6 internal vertices.

e  e′  e′′

Case II. e and e′ are connected to the same internal vertex v of D, yet some other edge e′′ exists in D that connects the skeleton of D to some other internal vertex v′ in D. In that case, use the previous case and the transitivity of equality: (elimination starting with e)=(elimination starting with e′′)=(elimination starting with e′).

24“7” here is a symbol that denotes an arbitrary natural number greater than 1 and “8” denotes 7 + 1.

⋯ k w_{k} = D_{L} =  D_{R} =

Figure 13. The left-arrow diagram D_{L}, the right-arrow diagram D_{R} and the k-wheel w_{k}.

ef  e′

Case III. Case III is what remains if neither Case I nor Case II hold. In that case, D must have a schematic form as on the right, with the “blob” not connected to the skeleton other than via e or e′, yet further arrows may exist outside of the blob. Let f denote the edge connecting the blob to e and e′. The “two in one out” rule for vertices implies that any part of a diagram must have an excess of incoming edges over outgoing edges, equal to the total number of vertices in that diagram part. Applying this principle to the blob, we find that it must contain exactly one vertex, and that f and therefore e and e′ must all be oriented upwards.

e f e′

We leave it to the reader to verify that in this case the two ways of applying the elimination procedure, e and then f or e′ and then f, yield the same answer modulo \overrightarrow{4T} (in fact, that answer is 0).

We also leave it to the reader to verify that \overrightarrow{ST U}_{1} implies \overrightarrow{AS} and \overrightarrow{IHX}. Algebraically, these are restatements of the anti-symmetry of the bracket and of Jacobi’s identity: if [x, y] := xy − yx, then 0 = [x, y] + [y, x] and [x, [y, z]] = [[x, y], z] − [[x, z], y]. □

Note that A^{wt}(↑) inherits algebraic structure from A^{w}(↑): it is an algebra by concatenation of diagrams, and a co-algebra with Δ(D), for D ∈ D^{wt}(↑), being the sum of all ways of dividing D between a “left co-factor” and a “right co-factor” so that connected components of D−S are kept intact, where S is the skeleton line of D (compare with [BN1, Definition 3.7]).

As A^{w}(↑) and A^{wt}(↑) are canonically isomorphic, from this point on we will not keep the distinction between the two spaces.

One may add the RI relation to the definition of A^{wt}(↑) to get a space A^{swt}(↑), or the FI relation to get A^{rwt}(↑). The statement and proof of the bracket rise theorem adapt with no difficulty, and we find that A^{sw}(↑) ≅ A^{swt}(↑) and A^{rw}(↑) ≅ A^{rwt}(↑).

Theorem 3.16. The bi-algebra A^{w}(↑) is the bi-algebra of polynomials in the diagrams D_{L}, D_{R} and w_{k} (for k ≥ 1) shown in Figure 13, where deg D_{L} = deg D_{R} = 1 and deg w_{k} = k, subject to the one relation w_{1} = D_{L} − D_{R}. Thus A^{w}(↑) has two generators in degree 1 and one generator in every degree greater than 1, as stated in Section 3.3.

Proof. (sketch). Readers familiar with the diagrammatic PBW theorem [BN1, Theorem 8] will note that it has an obvious analogue for the A^{w}(↑) case, and that the proof in [BN1] carries through almost verbatim. Namely, the space A^{w}(↑) is isomorphic to a space B^{w} of “unitrivalent diagrams” with symmetrized univalent ends modulo \overrightarrow{AS} and \overrightarrow{IHX}. Given the “two in one out” rule for arrow diagrams in A^{w}(↑) (and hence in B^{w}) the connected components of diagrams in B^{w} can only be trees or wheels. Trees vanish if they have more than one leaf, as their leafs are symmetric while their internal vertices are anti-symmetric, so B^{w} is generated by wheels (which become the w_{k}’s in A^{w}(↑)) and by the one-leaf-one-root tree, which is simply a single arrow, and which becomes the average of D_{L} and D_{R}. The relation w_{1} = D_{L} − D_{R} is then easily verified using \overrightarrow{ST U}_{2}.

One may also argue directly, without using sophisticated tools. In short, let D be a diagram in A^{w}(↑) and S is its skeleton. Then D − S may have several connected components, whose “legs” are intermingled along S. Using the \overrightarrow{ST U} relations these legs can be sorted (at a cost of diagrams with fewer connected components, which could have been treated earlier in an inductive proof). At the end of the sorting procedure one can see that the only diagrams that remain are our declared generators. It remains to show that our generators are linearly independent (apart for the relation w_{1} = D_{L} − D_{R}). For the generators in degree 1, simply write everything out explicitly in the spirit of Section 4.2.2. In higher degrees there is only one primitive diagram in each degree, so it is enough to show that w_{k} ≠ 0 for every k. This can be done “by hand”, but it is more easily done using Lie algebraic tools in Section 3.6. □

Exercise 3.17. Show that the bi-algebra A^{rw}(↑) (see Section 3.3) is the bi-algebra of polynomials in the wheel diagrams w_{k} (k ≥ 2), and that A^{sw}(↑) is the bi-algebra of polynomials in the same wheel diagrams and an additional generator D_{A} := D_{L} = D_{R}.

Theorem 3.18. In A^{w}(◯) all wheels vanish and hence the bi-algebra A^{w}(◯) is the bialgebra of polynomials in a single variable D_{L} = D_{R}.

Proof. This is Lemma 2.7 of [Na]. In short, a wheel in A^{w}(◯) can be reduced using \overrightarrow{ST U}_{2} to a difference of trees. One of these trees has two adjoining leafs and hence is 0 by TC and \overrightarrow{AS}. In the other two of the leafs can be commuted “around the circle” using TC until they are adjoining and hence vanish by TC and \overrightarrow{AS}. A picture is worth a thousand words, but sometimes it takes up more space. □

Exercise 3.19. Show that A^{sw}(◯) ≅ A^{w}(◯) yet A^{rw}(◯) vanishes except in degree 0.

The following two exercises may help the reader to develop a better “feel” for A^{w}(↑) and will be needed, within the discussion of the Alexander polynomial (especially within Definition 3.32).

= 0

Exercise 3.20. Show that the “Commutators Commute” (CC) relation, shown on the right, holds in A^{w}(↑). (Interpreted in Lie algebras as in the next section, this relation becomes [[x, y], [z, w]] = 0, and hence the name “Commutators Commute”). Note that the proof of CC depends on the skeleton having a single component; later, when we will work with A^{w}-spaces with more complicated skeleta, the CC relation will not hold.

h a i r Y  W

Exercise 3.21. Show that “detached wheels” and “hairy Y ’s” make sense in A^{w}(↑). As on the right, a detached wheel is a wheel with a number of spokes, and a hairy Y is a combinatorial Y shape with further “hair” on its trunk (its outgoing arrow). It is specified where the trunk and the leafs of the Y connect to the skeleton, but it is not specified where the spokes of the wheel and where the hair on the Y connect to the skeleton. The content of the exercise is to show that modulo the relations of A^{w}(↑), it is not necessary to specify this further information: all ways of connecting the spokes and the hair to the skeleton are equivalent. Like the previous exercise, this result depends on the skeleton having a single component.

Remark 3.22. In the case of classical knots and classical chord diagrams, Jacobi diagrams have a topological interpretation using the Goussarov-Habiro calculus of claspers [Gou2, Hab]. In the w case a similar such calculus was developed by Watanabe in [Wa]. Various related results are at [HKS, HS].

3.6. The Relation with Lie Algebras. The theory of finite type invariants of knots is related to the theory of metrized Lie algebras via the space A of chord diagrams, as explained in [BN1, Theorem 4, Exercise 5.1]. In a similar manner the theory of finite type invariants of w-knots is related to arbitrary finite-dimensional Lie algebras (or equivalently, to doubles of co-commutative Lie bialgebra) via the space A^{w}(↑) of arrow diagrams.

3.6.1. Preliminaries. Given a finite dimensional Lie algebra 𝔤 let I𝔤 := 𝔤^{∗} ⋊ 𝔤 be the semidirect product of the dual 𝔤^{∗} of 𝔤 with 𝔤, with 𝔤^{∗} taken as an Abelian algebra and with 𝔤 acting on 𝔤^{∗} by the usual coadjoint action. In formulae,

I𝔤 = {(φ, x): φ ∈ 𝔤^{∗}, x ∈ 𝔤},
[(φ_{1}, x_{1}), (φ_{2}, x_{2})] = (x_{1}φ_{2} − x_{2}φ_{1}, [x_{1}, x_{2}]).

In the case where 𝔤 is the algebra so(3) of infinitesimal symmetries of ℝ^{3}, its dual 𝔤^{∗} is itself ℝ^{3} with the usual action of so(3) on it, and I𝔤 is the algebra ℝ^{3} ⋊ so(3) of infinitesimal affine isometries of ℝ^{3}. This is the Lie algebra of the Euclidean group of isometries of ℝ^{3}, which is often denoted ISO(3). This explains our choice of the name I𝔤.

Note that if 𝔤 is a co-commutative Lie bialgebra then I𝔤 is the “double” of 𝔤 [Dr1]. This is a significant observation, for it is a part of the relationship between this paper and the Etingof-Kazhdan theory of quantization of Lie bialgebras [EK]. Yet we will make no explicit use of this observation below.

In the construction that follows we are going to construct a map from A^{w} to U(I𝔤), the universal enveloping algebra of I𝔤. Note that a map A^{w} → U(I𝔤) is “almost the same” as a map A^{sw} → U(I𝔤), in the following sense. There is an obvious quotient map p: A^{w} → A^{sw}, and p has a one-sided inverse F: A^{sw} → A^{w} defined by

F (D) = ∑_{k=0}^{∞} \frac{(−1)^{k}}{k!} S_{L}^{k}(D) · w_{1}^{k}.

Here S_{L} denotes the map that sends an arrow diagram to the sum of all ways of deleting a left-going arrow, and w_{1} denotes the 1-wheel, as shown in Figure 13. The reader can verify that F is well-defined, an algebra- and co-algebra homomorphism, and that p ∘ F = id_{A^{sw}}.

3.6.2. The Construction. Fixing a finite dimensional Lie algebra 𝔤 we construct a map T_{𝔤}^{w}: A^{w} → U(I𝔤) which assigns to every arrow diagram D an element of the universal enveloping algebra U(I𝔤). As is often the case in our subject, a picture of a typical example is worth more than a formal definition:

𝔤^{∗} ⊗ 𝔤^{∗} ⊗ 𝔤 ⊗ 𝔤 ⊗ 𝔤^{∗} ⊗ 𝔤^{∗}  U(I𝔤)^{I}^{𝔤^{∗}}  ^{𝔤^{∗}}  ^{𝔤}  ^{𝔤^{∗}}^{𝔤^{∗}}^{𝔤}^{B}  ^{B}^{contract}^{𝔤} ^{𝔤^{∗}}

In short, we break up the diagram D into its constituent pieces and assign a copy of the structure constants tensor B ∈ 𝔤^{∗} ⊗ 𝔤^{∗} ⊗ 𝔤 to each internal vertex v of D (keeping an association between the tensor factors in 𝔤^{∗} ⊗ 𝔤^{∗} ⊗ 𝔤 and the edges emanating from v, as dictated by the orientations of the edges and of the vertex v itself). We assign the identity tensor in 𝔤^{∗} ⊗ 𝔤 to every arrow in D that is not connected to an internal vertex, and contract any pair of factors connected by a fully internal arrow. The remaining tensor factors (𝔤^{∗} ⊗ 𝔤^{∗} ⊗ 𝔤 ⊗ 𝔤 ⊗ 𝔤^{∗} ⊗ 𝔤^{∗} in our examples) are all along the skeleton and can thus be ordered by the skeleton. We then multiply these factors to get an output T_{𝔤}^{w}(D) in U(I𝔤).

It is also useful to restate this construction given a choice of a basis. Let (x_{j}) be a basis

of 𝔤 and let (φ^{i}) be the dual basis of 𝔤^{∗}, so that φ^{i}(x_{j}) = δ_{j}^{i}, and let b_{ij}^{k} denote the structure constants of 𝔤 in the chosen basis: [x_{i}, x_{j}] = ∑ b_{ij}^{k}x_{k}. Mark every arrow in D with lower case Latin letter from within {i, j, k, … }^{25}. Form a product P_{D} by taking one b_{αβ}^{γ} factor for each internal vertex v of D using the letters marking the edges around v for α, β and γ and by taking one x_{α} or φ^{β} factor for each skeleton vertex of D, taken in the order that they appear along the skeleton, with the indices α and β dictated by the edge markings and with the choice between factors in 𝔤 and factors in 𝔤^{∗} dictated by the orientations of the edges. Finally let T_{𝔤}^{w}(D) be the sum of P_{D} over the indices i, j, k, … running from 1 to dim 𝔤:

i  ^{j}  ^{k}  ^{l}^{m}^{n}  ^{b_{kl}^{m}}^{b_{ji}^{k}}  ^{∑_{i,j,k,l,m,n=1}^{dim 𝔤} b_{ij}^{k}b_{kl}^{m}φ^{i}φ^{j}x_{n}x_{m}φ^{l} ∈ U(I𝔤)}^{φ^{i}}  ^{φ^{j}}  ^{x_{n}} ^{x_{m}} ^{φ^{n}}  ^{φ^{l}}  (19)

The following is easy to verify (compare with [BN1, Theorem 4, Exercise 5.1]):

Proposition 3.23. The above two definitions of T_{𝔤}^{w} agree, are independent of the choices made within them, and respect all the relations defining A^{w}. □

While we do not provide a proof of this proposition here, it is worthwhile to state the correspondence between the relations defining A^{w} and the Lie algebraic information in U(I𝔤): \overrightarrow{AS} is the antisymmetry of the bracket of 𝔤, \overrightarrow{IHX} is the Jacobi identity of 𝔤, \overrightarrow{ST U}_{1} and \overrightarrow{ST U}_{2} are the relations [x_{i}, x_{j}] = x_{i}x_{j} − x_{j}x_{i} and [φ^{i}, x_{j}] = φ^{i}x_{j} − x_{j}φ^{i} in U(I𝔤), T C is the fact that 𝔤^{⋆} is taken as an Abelian algebra, and \overrightarrow{4T} is the fact that the identity tensor in 𝔤^{∗} ⊗ 𝔤 is 𝔤-invariant.

3.6.3. Example: The 2 Dimensional Non-Abelian Lie Algebra. Let 𝔤 be the Lie algebra with two generators x_{1,2} satisfying [x_{1}, x_{2}] = x_{2}, so that the only non-vanishing structure constants b_{ij}^{k} of 𝔤 are b_{12}^{2} = −b_{21}^{2} = 1. Let φ^{i} ∈ 𝔤^{∗} be the dual basis of x_{i}; by an easy calculation, we find that in I𝔤 the element φ^{1} is central, while [x_{1}, φ^{2}] = −φ^{2} and [x_{2}, φ^{2}] = φ^{1}. We calculate T_{𝔤}^{w}(D_{L}), T_{𝔤}^{w}(D_{R}) and T_{𝔤}^{w}(w_{k}) using the “in basis” technique of Equation (19). The outputs of these calculations lie in U(I𝔤); we display these results in a PBW basis in which the elements of 𝔤^{∗} precede the elements of 𝔤:

25The supply of these can be made inexhaustible by the addition of numerical subscripts.

Figure 14. A long 8_{17}, with the span of crossing #3 marked. The projection is as in Brian Sanderson’s garden. See [BND2]/SandersonsGarden.html.

T_{𝔤}^{w}(D_{L}) = x_{1}φ^{1} + x_{2}φ^{2} = φ^{1}x_{1} + φ^{2}x_{2} + [x_{2}, φ^{2}] = φ^{1}x_{1} + φ^{2}x_{2} + φ_{1},
T_{𝔤}^{w}(D_{R}) = φ^{1}x_{1} + φ^{2}x_{2},  (20)
T_{𝔤}^{w}(w_{k}) = (φ^{1})^{k}.

1  1  1  1 2 2 2 2 φ^{1} φ^{1} φ^{1} φ^{1}

For the last assertion above, note that all non-vanishing structure constants b_{ij}^{k} in our case have k = 2, and therefore all indices corresponding to edges that exit an internal vertex must be set equal to

2. This forces the “hub” of w_{k} to be marked 2 and therefore the legs to be marked 1, and therefore w_{k} is mapped to (φ^{1})^{k}.

Note that the calculations in (20) are consistent with the relation D_{L} − D_{R} = w_{1} of Theorem 3.16 and that they show that other than that relation, the generators of A^{w} are linearly independent.

3.7. The Alexander Polynomial. Let K be a long w-knot, let Z(K) be the invariant of Theorem 3.11. Theorem 3.27 below asserts that apart from self-linking, Z(K) contains precisely the same information as the Alexander polynomial A(K) of K (defined below). But we have to start with some definitions.

Definition 3.24. Enumerate the crossings of K from 1 to n in some arbitrary order. For 1 ≤ j ≤ n, the “span” of crossing #i is the connected open interval along the line parametrizing K between the two times K “visits” crossing #i (see Figure 14). Form a matrix T = T (K) with T_{ij} the indicator function of “the lower strand of crossing #j is within the span of crossing #i” (so T_{ij} is 1 if for a given i, j the quoted statement is true, and 0 otherwise). Let s_{i} be the sign of crossing #i ((−, −, −, −, +, +, +, +) for Figure 14), let d_{i} be +1 if K visits the “over” strand of crossing #i before visiting the “under” strand of that crossing, and let d_{i} = −1 otherwise ((−, +, −, +, −, +, −, +) for Figure 14). Let S = S(K) be the diagonal matrix with S_{ii} = s_{i}d_{i}, and for an indeterminate X, let X^{−S} denote the diagonal matrix with diagonal entries X^{−s_{i}d_{i}}. Finally, let A(K) be the Laurent polynomial in ℤ[X, X^{−1}] given by

A(K)(X) := det (I + T (I − X^{−S})) .  (21)

Example 3.25. For the knot diagram in Figure 14,

T =( \begin{matrix}0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \end{matrix} ), S=( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ), and A=| \begin{matrix}1 & 1−X & 1−X^{−1} & 1−X & 1−X & 0 & 1−X & 0 \\ 0 & 1 & 1−X^{−1} & 0 & 1−X & 0 & 0 & 0 \\ 0 & 1−X & 1 & 0 & 1−X & 0 & 0 & 0 \\ 0 & 1−X & 0 & 1 & 1−X & 0 & 1−X & 0 \\ 0 & 1−X & 0 & 1−X & 1 & 1−X^{−1} & 1−X & 1−X^{−1} \\ 0 & 1−X & 0 & 1−X & 0 & 1 & 1−X & 0 \\ 0 & 0 & 0 & 1−X & 0 & 1−X^{−1} & 1 & 0 \\ 0 & 0 & 0 & 1−X & 0 & 1−X^{−1} & 0 & 1 \end{matrix} |.

The last determinant equals −X^{3} + 4X^{2} − 8X + 11 − 8X^{−1} + 4X^{−2} − X^{−3}, the Alexander polynomial of the knot 8_{17} (e.g. [Rol]).

Theorem 3.26. (P. Lee, [Lee2]) For any (classical) knot K, A(K) is equal to the normalized Alexander polynomial [Rol] of K. □

The Mathematica notebook [BND2, “wA”] verifies Theorem 3.26 for all prime knots with up to 11 crossings.

The following theorem asserts that Z(K) can be computed from A(K) (Equation (22)) and that modulo a certain additional relation and with the appropriate identifications in place, Z(K) is A(K) (Equation (23)).

Theorem 3.27. (Proof in Section 3.8). Let x be an indeterminate, let sl be self-linking as in Exercise 3.6, let D_{A} := D_{L} = D_{R} and w_{k} be as in Figure 13, and let w: ℚ⟦x⟧ → A^{w} be the linear map defined by x^{k} ↦ w_{k}. Then for a w-knot K,

Z(K) = \underbrace{exp_{A^{sw}} (sl(K)D_{A})}_{sl coded in arrows} · \underbrace{exp_{A^{sw}} (−w (log_{ℚ⟦x⟧} A(K)(e^{x})))}_{main part: Alexander coded in wheels},  (22)

where the logarithm and inner exponentiation are computed by formal power series in ℚ⟦x⟧ and the outer exponentiations are likewise computed in A^{sw}.

= w^{2} · w^{3}  w^{5}

Let A^{reduced} be A^{sw} modulo the additional relations D_{A} = 0 and w_{k}w_{l} = w_{k+l} for k, l ≠ 1. The quotient A^{reduced} can be identified with vector space of (infinite) linear combinations of w_{k}’s (with k ≠ 1). Identifying the k-wheel w_{k} with x^{k}, we see that A^{reduced} is the space of power series in x having no linear terms. Note by inspecting (21) that A(K)(e^{x}) never has a term linear in x, and that modulo w_{k}w_{l} = w_{k+l}, the exponential and the logarithm in (22) cancel each other out. Hence within A^{reduced},

Z(K) = A^{−1}(K)(e^{x}).  (23)

Remark 3.28. In [HKS] K. Habiro, T. Kanenobu, and A. Shima show that all coefficients of the Alexander polynomial are finite type invariants of w-knots, and in [HS] K. Habiro and A. Shima show that all finite type invariants of w-knots are polynomials in the coefficients of the Alexander polynomial. Thus Theorem 3.27 is merely an “explicit form” of these earlier results.

3.8. Proof of Theorem 3.27. We start with a sketch. The proof of Theorem 3.27 can be divided in three parts: differentiation, bulk management, and computation.

Differentiation. Both sides of our goal, Equation (22), are exponential in nature. When seeking to show an equality of exponentials it is often beneficial to compare their derivatives^{26}. In our case the useful “derivatives” to use are the “Euler operator” E (“multiply every term by its degree”, an analogue of f ↦ xf′, defined in Section 3.8.1), and the “normalized Euler operator” Z ↦ ẼZ := Z^{−1}EZ, which is a variant of the logarithmic derivative f ↦ x(log f)′ = xf′/f. Since Ẽ is one to one (Section 3.8.1) and since we know how to apply Ẽ to the right hand side of Equation (22) (Section 3.8.1), it is enough to show that with B := T (exp(−xS) − I) and suppressing the fixed w-knot K from the notation,

EZ = Z · (sl · D_{A} − w[x tr ((I − B)^{−1}T S exp(−xS))])  in A^{sw}.  (24)

Bulk Management. Next we seek to understand the left hand side of (24). Z is made up of “quantities in bulk”: arrows that come in exponential “reservoirs”. As it turns out, EZ is made up of the same bulk quantities, but also allowing for a single non-bulk “red excitation” (compare with Ee^{x} = x · e^{x}; the “bulk” e^{x} remains, and single “excited red” x gets created). We wish manipulate and simplify that red excitation. This is best done by introducing a certain module, IAM_{K}, the “Infinitesimal Alexander Module” of K (see Section 3.8.2). The elements of IAM_{K} can be thought of as names for “bulk objects with a red excitation”, and hence there is an “interpretation map” ι: IAM_{K} → A^{sw}, which maps every “name” into the object it represents. There are three special elements in IAM_{K}: an element λ, which is the name of EZ (that is, ι(λ) = EZ), the element δ_{A} which is the name of D_{A} · Z (so ι(δ_{A}) = D_{A} · Z), and an element ω_{1} which is the name of a “detached” 1-wheel that is appended to Z. The latter can take a coefficient which is a power of x, with ι(x^{k}ω_{1}) = w(x^{k+1})·Z = (Z times a (k +1)-wheel). Thus it is enough to show that in IAM_{K},

λ = sl · δ_{A} − tr ((I − B)^{−1}T SX^{−S}) ω_{1}, with X = e^{x}.  (25)

Indeed, applying ι to both sides of the above equation, we get Equation (24) back again. Computation. Last, we show in Section 3.8.3 that (25) holds true. This is a computation that happens entirely in IAM_{K} and does not mention finite type invariants, expansions or arrow diagrams in any way.

3.8.1. The Euler Operator. Let A be a completed graded algebra with unit, in which all degrees are ≥ 0. Define a continuous linear operator E: A → A by setting Ea = (deg a)a for homogeneous a ∈ A. In the case A = ℚ⟦x⟧, we have Ef = xf′, the standard “Euler operator”, and hence we adopt this name for E in general.

We say that Z ∈ A is a “perturbation of the identity” if its degree 0 piece is 1. Such a Z is always invertible. For such a Z, set ẼZ := Z^{−1} · EZ, and call the thus (partially) defined operator Ẽ: A → A the “normalized Euler operator”. From this point on when we write ẼZ for some Z ∈ A, we automatically assume that Z is a perturbation of the identity or that it is trivial to show that Z is a perturbation of the identity. Note that for f ∈ ℚ⟦x⟧, we have Ẽf = x(log f)′, so Ẽ is a variant of the logarithmic derivative.

Claim 3.29. Ẽ is one to one.

26Thanks, Dylan.

Proof. Assume Z_{1} ≠ Z_{2} and let d be the smallest degree in which they differ. Then d > 0 and in degree d the difference ẼZ_{1} − ẼZ_{2} is d times the difference Z_{1} − Z_{2}, and hence ẼZ_{1} ≠ ẼZ_{2}. □

Thus in order to prove our goal, Equation (22), it is enough to compute Ẽ of both sides and to show the equality then. We start with the right hand side of (22); but first, we need some simple properties of E and Ẽ. The proofs of these properties are routine and hence they are omitted.

Proposition 3.30. The following hold true:

(1) E is a derivation: E(fg) = (Ef)g + f(Eg).

(2) If Z_{1} commutes with Z_{2}, then Ẽ(Z_{1}Z_{2}) = ẼZ_{1} + ẼZ_{2}.

(3) If z commutes with Ez, then Ee^{z} = e^{z}(Ez) and Ẽe^{z} = Ez.

(4) If w: A → A is a morphism of graded algebras, then it commutes with E and Ẽ. □

Let us denote the right hand side of (22) by Z_{1}(K). Then by the above proposition, remembering (Theorem 3.16) that A^{sw} is commutative and that deg D_{A} = 1, we have

ẼZ_{1}(K) = sl · D_{A} − w(E log A(K)(e^{x})) = sl · D_{A} − w (x\frac{d}{dx} log A(K)(e^{x})) .

The rest is an exercise in matrices and differentiation. A(K) is a determinant (21), and in general, \frac{d}{dx} log det(M) = tr (M^{−1} \frac{d}{dx}M). So with B = T (e^{−xS} − I) (so M = I − B), we have

ẼZ_{1}(K) = sl · D_{A} + w (x tr ((I − B)^{−1} \frac{d}{dx}B)) = sl · D_{A} − w (x tr ((I − B)^{−1}T Se^{−xS})) ,

as promised in Equation (24).

3.8.2. The Infinitesimal Alexander Module. Let K be a w-knot diagram. The Infinitesimal Alexander Module IAM_{K} of K is a certain module made from a certain space IAM _{K}^{0} of pictures “annotating” K with “red excitations” modulo some pictorial relations that indicate how the red excitations can be moved around. The space IAM _{K}^{0} in itself is made of three pieces, or “sectors”. The “A sector” in which the excitations are red arrows, the “Y sector” in which the excitations are “red hairy Y-diagrams”, and a rank 1 “W sector” for “red hairy wheels”. There is an “interpretation map” ι: IAM _{K}^{0} → A^{w} which descends to a well defined (and homonymous) ι: IAM_{K} → A^{w}. Finally, there are some special elements λ and δ_{A} that live in the A sector of IAM _{K}^{0} and ω_{1} that lives in the W sector.

In principle, the description of IAM _{K}^{0} and of IAM_{K} can be given independently of the interpretation map ι, and there are some good questions to ask about IAM_{K} (and the special elements in it) that are completely independent of the interpretation of the elements of IAM_{K} as “perturbed bulk quantities” within A^{sw}. Yet IAM_{K} is a complicated object and we fear its definition will appear completely artificial without its interpretation. Hence below the two definitions will be woven together.

IAM_{K} and ι may equally well be described in terms of K or in terms of the Gauss diagram of K (Remark 3.4). For pictorial simplicity, we choose to use the latter; so let G = G(K) be the Gauss diagram of K. It is best to read the following definition while at the same time studying Figure 15.

−  +  −  +  −  +  −  + red  red K  G  in A  in Y  in W red

Figure 15. A sample w-knot K, it’s Gauss diagram G, and one generator from each of the A, Y, and W sectors of IAM _{K}^{0}. Red parts are marked with the word “red”.

Definition 3.31. Let R be the ring ℤ[X, X^{−1}] of Laurent polynomials in X, and let R_{1} be the subring of polynomials that vanish at X = 1 (i.e., whose sum of coefficients is 0)^{27}. Let IAM _{K}^{0} be the direct sum of the following three modules (which for the purpose of taking the direct sum, are all regarded as ℤ-modules):

(1) The “A sector” is the free ℤ-module generated by all diagrams made from G by the addition of a single unmarked “red excitation” arrow, whose endpoints are on the skeleton of G and are distinct from each other and from all other endpoints of arrows in G. Such diagrams are considered combinatorially — so two are equivalent iff they differ only by an orientation preserving diffeomorphism of the skeleton. Let us count: if K has n crossings, then G has n arrows and the skeleton of G get subdivided into m := 2n + 1 arcs. An A sector diagram is specified by the choice of an arc for the tail of the red arrow and an arc for the head (m^{2} choices), except if the head and the tail fall within the same arc, their relative ordering has to be specified as well (m further choices). So the rank of the A sector over ℤ is m(m + 1).

(2) The “Y sector” is the free R_{1}-module generated by all diagrams made from G by the addition of a single “red excitation” Y -shape single-vertex graph, with two incoming edges (“tails”) and one outgoing (“head”), modulo anti-symmetry for the two incoming edges (again, considered combinatorially). Counting is more elaborate: when the three edges of the Y end in distinct arcs in the skeleton of G, we have \frac{1}{2}m(m − 1)(m − 2) possibilities (\frac{1}{2} for the antisymmetry). When the two tails of the Y lie on the same arc, we get 0 by anti-symmetry. The remaining possibility is to have the head and one tail on one arc (order matters!) and the other tail on another, at 2m(m − 1) possibilities. So the rank of the Y sector over R_{1} is m(m − 1)(\frac{1}{2}m + 1).

(3) The “W sector” is the rank 1 free R-module with a single generator w_{1}. It is not necessary for w_{1} to have a pictorial representation, yet one, involving a single “red” 1-wheel, is shown in Figure 15.

Definition 3.32. The “interpretation map” ι: IAM _{K}^{0} → A^{w} is defined by sending the arrows (marked + or −) of a diagram in IAM _{K}^{0} to e^{±a}-exponential reservoirs of arrows, as in the definition of Z (see Remark 3.12). In addition, the red excitations of diagrams in IAM _{K}^{0} are interpreted as follows:

(1) In the A sector, the red arrow is simply mapped to itself, with the colour red suppressed.

(2) In the Y sector diagrams have red Y ’s and coefficients f ∈ R_{1}. Substitute X = e^{x} in f, expand in powers of x, and interpret x^{k}Y as a “hairy Y with k − 1 hairs” as in

27R_{1} is only very lightly needed, and only within Definition 3.32. In particular, all that we say about IAM_{K} that does not concern the interpretation map ι is equally valid with R replacing R_{1}.

r  r  s  s s  s r  r r  r  r r  r r  r  r  r  r r r − r r  =  = 0 A_{tt}  = 0 s − s  = 0 ss rr  r −  s s r s r  = − s  (X^{s} − 1) = s − ss  s r r A_{ht}  = s −  s s  r  (X^{s} − 1)  = s − s  s (1 − X^{s}) Y_{th} Y_{tt} Y_{ht}  Y_{hh} A_{th} A_{hh} A_{w}  Y_{w}  (1 − X^{s}) (X^{s} − 1)  (1 − X^{s}) −  = r r

Figure 16. The relations R making IAM_{K}.

Exercise 3.21. Note that f(1) = 0, so only positive powers of x occur, so we never need to worry about “Y ’s with −1 hairs”. This is the only point where the condition f ∈ R_{1} (as opposed to f ∈ R) is needed.

(3) In the W sector treat the coefficients as above, but interpret x^{k}w_{1} as a detached w_{k+1}. I.e., as a detached wheel with k + 1 spokes, as in Exercise 3.21.

As stated above, IAM_{K} is the quotient of IAM _{K}^{0} by some set of relations. The best way to think of this set of relations is as “everything that’s obviously annihilated by ι”. Here’s the same thing, in a more formal language:

Definition 3.33. Let IAM_{K} := IAM _{K}^{0}/R, where R is the linear span of the relations depicted in Figure 16. The top 8 relations are about moving a leg of the red excitation across an arrow head or an arrow tail in G. Since the red excitation may be either an arrow (A) or a Y, its leg in motion may be either a tail or a head, and it may be moving either past a tail or past a head, there are 8 relations of that type. The next relation corresponds to D_{L} − D_{R} = w_{1} = 0. The last relation indicates the “price” (always a red w_{1}) of commuting a red head across a red tail. As per custom, in each case only the changing part of the diagrams involved is shown. Further, the red excitations are marked with the letter “r” and the sign of an arrow in G is marked s; so always s ∈ {±1}. The relations in the left column may be multiplied by a scalar in ℤ, while the relations in the right column may be multiplied by a scalar in R. Hence, for example, x^{0}w_{1} = 0 by A_{w}, yet x^{k}w_{1} ≠ 0 for k > 0.

Proposition 3.34. The interpretation map ι indeed annihilates all the relations in R.

Proof. ιA_{tt} and ιY_{tt} follow immediately from “Tails Commute”. The formal identity e^{ad b}(a) = e^{b}ae^{−b} implies e^{ad b}(a)e^{b} = e^{b}a and hence ae^{b} − e^{b}a = (1 − e^{ad b})(a)e^{b}. With a interpreted as “red head”, b as “black head”, and ad b as “hair” (justified by the ι-meaning of hair and by the \overrightarrow{ST U}_{1} relation, Figure 11), the last equality becomes a proof of ιY_{hh}.

+  +  −  +  +  −  +  +  − r  r  r λ =  +  − ω_{1} = +  +  −  r  +  +  − r δ_{R} =

Figure 17. The special elements ω_{1}, δ_{A}, and λ in IAM_{G}, for a sample 3-arrow Gauss diagram G.

Further pushing that same equality, we get ae^{b} − e^{b}a = \frac{1−e^{ad b}}{ad b} ([b, a]), where \frac{1−e^{ad b}}{ad b} is first interpreted as a power series \frac{1−e^{y}}{y} involving only non-negative powers of y, and then the substitution y = ad b is made. But that’s ιA_{hh}, when one remembers that ι on the Y sector automatically contains a single “\frac{1}{hair}” factor. Similar arguments, though using \overrightarrow{ST U}_{2} instead of \overrightarrow{ST U}_{1}, prove that Y_{ht}, Y_{th}, A_{ht}, and A_{th} are all in ker ι. Finally, ιA_{w} is RI, and ιY_{w} is a direct consequence of \overrightarrow{ST U}_{2}. □

Finally, we come to the special elements λ, δ_{A}, and ω_{1}.

Definition 3.35. Within IAM_{G}, let ω_{1} be, as before, the generator of the W sector. Let δ_{A} be a “short” red arrow, as in the A_{w} relation (exercise: modulo R, this is independent of the placement of the short arrows within G). Finally, let λ be the signed sum of exciting each of the (black) arrows in G in turn. The picture says all, and it is Figure 17.

Proposition 3.36. In A^{sw}(↑), the special elements of IAM_{G} are interpreted as follows: ι(ω_{1}) = Zw_{1}, ι(δ_{A}) = ZD_{A}, and most interesting, ι(λ) = EZ. Therefore, Equation (25) (if true) implies Equation (24) and hence it implies our goal, Theorem 3.27.

Proof. For the proof of this proposition, the only thing that isn’t done yet and isn’t trivial is the assertion ι(λ) = EZ. But this assertion is a consequence of Ee^{±a} = ±ae^{±a} and of a Leibniz law for the derivation E, appropriately generalized to a context where Z can be thought of as a “product” of “arrow reservoirs”. The details are left to the reader. □

3.8.3. The Computation of λ. Naturally, our next task is to prove Equation (25). This is done entirely algebraically within the finite rank module IAM_{G}. To read this section one need not know about A^{sw}(↑), or ι, or Z, but we do need to lay out some notation. Start by marking the arrows of G with a_{1} through a_{n} in some order.

Let ϵ stand for the informal yet useful quantity “a little”. Let λ_{ij} denote the difference λ′_{ij} − λ′′_{ij} of red excitations in the A sector of IAM_{G}, where λ′_{ij} is the diagram with a red arrow whose tail is ϵ to the right of the left end of a_{i} and whose head is \frac{1}{2}ϵ away from head of a_{j} in the direction of the tail of a_{j}, and where λ′′_{ij} has a red arrow whose tail is ϵ to the left of the right end of a_{i} and whose head is as before, \frac{1}{2}ϵ away from head of a_{j} in the direction of the tail of a_{j}. Let Λ = (λ_{ij}) be the matrix whose entries are the λ_{ij}’s, as shown in Figure 18.

Similarly, let y_{ij} denote the element in the Y sector of IAM_{G} whose red Y has its head \frac{1}{2}ϵ away from head of a_{j} in the direction of the tail of a_{j}, its right tail (as seen from the head) ϵ to the left of the right end of a_{i} and its left tail ϵ to the right of the left end of a_{i}. Let Y = (y_{ij}) be the matrix whose entries are the y_{ij}’s, as shown in Figure 18.

Proposition 3.37. With S and T as in Definition 3.24, and with B = T (X^{−S} − I) and λ as above, the following identities between elements of IAM_{G} and matrices with entries in

−  −  Y1 2 j i Λ  1  2 1 2 j i  1  2 −  −

Figure 18. The matrices Λ and Y for a sample 2-arrow Gauss diagram (the signs on a_{1} and a_{2} are suppressed, and so are the r marks). The twists in y_{11} and y_{22} may be replaced by minus signs.

IAM_{G} hold true:

λ − sl · D_{A} = tr SΛ  (26)
Λ = −BY − T X^{−S}w_{1}  (27)
Y = BY + T X^{−S}w_{1}  (28)

Proof of Equation (25) given Proposition 3.37. The last of the equalities above implies that Y = (I − B)^{−1}T X^{−S}w_{1}. Thus

λ − sl · D_{A} = tr SΛ = − tr S(BY + T X^{−S}w_{1}) = − tr S(B(I − B)^{−1}T X^{−S} + T X^{−S})w_{1}
= − tr ((I − B)^{−1}T SX^{−S}) w_{1},

and this is exactly Equation (25). □ Proof of Proposition 3.37. Equation (26) is trivial. The proofs of Equations (27) and (28) both have the same simple cores, that have to be supplemented by highly unpleasant tracking of signs and conventions and powers of X. Let us start from the cores.

To prove Equation (27) we wish to “compute” λ_{ik} = λ′_{ik} − λ′′_{ik}. As λ′_{ik} and λ′′_{ik} have their heads in the same place, we can compute their difference by gradually sliding the tail of λ′_{ik} from its original position near the left end of a_{i} towards the right end of a_{i}, where it would be cancelled by λ′′_{ik}. As the tail slides we pick up a y_{jk} term each time it crosses a head of an a_{j} (relation A_{th}), we pick up a vanishing term each time it crosses a tail (relation A_{tt}), and we pick up a w_{1} term if the tail needs to cross over its own head (relation A_{w}). Ignoring signs and (X^{±1} − 1) factors, the sum of the y_{jk}-terms should be proportional to T Y, for indeed, the matrix T has non-zero entries precisely when the head of an a_{j} falls within the span of an a_{i}. Unignoring these signs and factors, we get −BY (recall that B = T (X^{−S} − I) is just T with added (X^{±1} − 1) factors). Similarly, a w_{1} term arises in this process when a tail has to cross over its own head, that is, when the head of a_{k} is within the span of a_{i}. Thus the w_{1} term should be proportional to T w_{1}, and we claim it is −T X^{−S}w_{1}.

The core of the proof of Equation (28) is more or less the same. We wish to “compute” y_{ik} by sliding its left leg, starting near the left end of a_{i}, towards its right leg, which is stationary near the right end of a_{i}. When the two legs come together, we get 0 because of the anti-symmetry of Y excitations. Along the way we pick up further Y terms from the Y_{th} relations, and sometimes a w_{1} term from the Y_{w} relation. When all signs and (X^{±1} − 1) factors are accounted for, we get Equation (28).

We leave it to the reader to complete the details in the above proofs. It is a major headache, and we would not have trusted ourselves had we not written a computer program to manipulate quantities in IAM_{G} by a brute force application of the relations in R. Everything checks; see [BND2, “The Infinitesimal Alexander Module”]. □

This concludes the proof of Theorem 3.27. □

Remark 3.38. We chose the name “Infinitesimal Alexander Module” as in our mind there is some similarity between IAM_{K} and the “Alexander Module” of K. Yet beyond the above, we did not embark on any serious study of IAM_{K}. In particular, we do not know if IAM_{K} in itself is an invariant of K (though we suspect it wouldn’t be hard to show that it is), we do not know if IAM_{K} contains any further information beyond sl and the Alexander polynomial, and we do not know if there is any formal relationship between IAM_{K} and the Alexander module of K.

Remark 3.39. The logarithmic derivative of the Alexander polynomial also appears in Lescop’s [Les1, Les2]. We don’t know if its appearances there are related to its appearance here.

3.9. The Relationship with u-Knots. Unlike in the case of braids, there is a canonical universal finite type invariant of u-knots: the Kontsevich integral Z^{u}. So it makes sense to ask how it is related to the expansion Z^{w}.

\xymatrix{K^{u}(↑) \ar[r]^{Z^{u}} \ar[d]^{a} & A^{u}(↑) \ar[d]^{α} \\ K^{w}(↑) \ar[r]^{Z^{w}} & A^{w}(↑)}

We claim that the square on the left commutes, where K^{u}(↑) stands for long u-knots (knottings of an oriented line), and similarly K^{w}(↑) denotes long w-knots. As before, a is the composition of the maps u-knots → v-knots → w-knots, and α is the induced map on the projectivizations, mapping each chord to the sum of the two ways to direct

it.

Recall that α kills everything but wheels and arrows. We are going to use the formula for the “wheel part” of the Kontsevich integral as stated in [Kr]. Let K be a 0-framed long knot, and let A(K) denote the Alexander polynomial. Then by [Kr],

Z^{u}(K) = exp_{A^{u}} (−\frac{1}{2} log A(K)(e^{h})|_{h^{2n}→w_{2n}^{u}}) + “loopy terms”,

where w_{2n}^{u} stands for the unoriented wheel with 2n spokes; and “loopy terms” means terms that contain diagrams with more than one loop, which are killed by α. Note that by the symmetry A(z) = A(z^{−1}) of the Alexander polynomial, A(K)(e^{h}) contains only even powers of h, as suggested by the formula.

We need to understand how α acts on wheels. Due to the two-in-one-out rule, a wheel is zero unless all the “spokes” are oriented inward, and the cycle oriented in one direction. In other words, there are two ways to orient an unoriented wheel: clockwise or counterclockwise. Due to the anti-symmetry of chord vertices, we get that for odd wheels α(w_{2h+1}^{u}) = 0 and for even wheels α(w_{2h}^{u}) = 2w_{2h}^{w}. As a result,

αZ^{u}(K) = exp_{A^{w}} (−\frac{1}{2} log A(K)(e^{h})|_{h^{2n}→2w_{2n}}) = exp_{A^{w}} (− log A(K)(e^{h})|_{h^{2n}→w_{2n}})

which agrees with the formula (22) of Theorem 3.27. Note that since K is 0-framed, the first part (“sl coded in arrows”) of (22) is trivial.

4. Odds and Ends

4.1. What means “closed form”? As stated earlier, one of my hopes for this paper is that it will lead to closed-form formulae for tree-level associators. The notion “closed-form” in itself requires an explanation (see footnote 3). Is e^{x} a closed form expression for ∑_{n=0}^{∞} \frac{x^{n}}{n!} , or is it just an artificial name given for a transcendental function we cannot otherwise reduce? Likewise, why not call some tree-level associator Φ^{tree} and now it is “in closed form”?

For us, “closed-form” should mean “useful for computations”. More precisely, it means that the quantity in question is an element of some space A^{cf} of “useful closed-form thingies” whose elements have finite descriptions (hopefully, finite and short) and on which some operations are defined by algorithms which terminate in finite time (hopefully, finite and short). Furthermore, there should be a finite-time algorithm to decide whether two descriptions of elements of A^{cf} describe the same element^{28}. It is even better if the said decision algorithm takes the form “bring each of the two elements in question to a canonical form by means of some finite (and hopefully short) procedure, and then compare the canonical forms verbatim”; if this is the case, many algorithms that involve managing a large number of elements become simpler and faster.

Thus for example, polynomials in a variable x are always of closed form, for they are simply described by finite sequences of integers (which in themselves are finite sequences of digits), the standard operations on polynomials (+, ×, and, say, \frac{d}{dx}) are algorithmically computable, and it is easy to write the “polynomial equality” computer program. Likewise for rational functions and even for rational functions of x and e^{x}.

On the other hand, general elements Φ of the space A^{tree}(↑_{3}) of potential tree-level associators are not closed-form, for they are determined by infinitely many coefficients. Thus iterative constructions of associators, such as the one in [BN3] are computationally useful only within bounded-degree quotients of A^{tree}(↑_{3}) and not as all-degree closed-form formulae. Likewise, “explicit” formulae for an associator Φ in terms of multiple ζ-values (e.g. [LM1]) are not useful for computations as it is not clear how to apply tangle-theoretic operations to Φ (such as Φ ↦ Φ^{1342} or Φ ↦ (1 ⊗ Δ ⊗ 1)Φ) while staying within some space of “objects with finite description in terms of multiple ζ-values”. And even if a reasonable space of such objects could be defined, it remains an open problem to decide whether a given rational linear combination of multiple ζ-values is equal to 0.

4.2. Arrow Diagrams to Degree 2. Just as an example, in this section we study the spaces A^{−}(↑), A^{s−}(↑), A^{r−}(↑), P^{−}(↑), A^{−}(◯), A^{s−}(◯), and A^{r−}(◯) in degrees m ≤ 2 in detail, both in the “v” case and in the “w” case (the “u” case has been known since long).

4.2.1. Arrow Diagrams in Degree 0. There is only one degree 0 arrow diagram, the empty diagram D_{0} (see Figure 19). There are no relations, and thus {D_{0}} is the basis of all G_{0}A^{−}(↑) spaces and its obvious closure, the empty circle, is the basis of all G_{0}A^{−}(◯) spaces. D_{0} is the unit 1, yet ΔD_{0} = D_{0} ⊗ D_{0} = 1 ⊗ 1 ≠ D_{0} ⊗ 1 + 1 ⊗ D_{0}, so D_{0} is not primitive and dim G_{0}P^{−}(↑) = 0.

28In our context, if it is hard to decide within the target space of an invariant whether two elements are equal or not, the invariant is not too useful in deciding whether two knotted objects are equal or not.

D_{0} = D_{R} = D^{L} =  D_{1} = D_{2} = D^{3} =  D_{4} = D_{5} = D^{6} =  D_{7} = D_{8} = D^{9} =  D_{10} = D_{11} = D^{12} =

Figure 19. The 15 arrow diagrams of degree at most 2.

4.2.2. Arrow Diagrams in Degree 1. There is only two degree 1 arrow diagrams, the “right arrow” diagram D_{R} and the “left arrow” diagram D_{L} (see Figure 19). There are no 6T relations, and thus {D_{R}, D_{L}} is the basis of G_{1}A^{−}(↑). Modulo RI, D_{L} = D_{R} and hence D_{A} := D_{L} = D_{R} is the single basis element of G_{1}A^{s−}(↑). Both D_{R} and D_{L} vanish modulo FI, so dim G_{1}A^{r−}(↑) = dim G_{1}A^{r−}(◯) = 0. Both D_{R} and D_{L} are primitive, so dim G_{1}P^{−}(↑) = 2. Finally, the closures D̄_{R} and D̄_{L} of D_{R} and D_{L} are equal, so G_{1}A^{s−}(◯) = G_{1}A^{−}(◯) = ⟨D̄_{R}⟩ = ⟨D̄_{L}⟩ = ⟨D̄_{A}⟩.

4.2.3. Arrow Diagrams in Degree 2. There are 12 degree 2 arrow diagrams, which we denote D_{1}, …, D_{12} (see Figure 19). There are six 6T relations, corresponding to the 6 ways of ordering the 3 vertical strands that appear in a 6T relation (see Figure 3) along a long line. The ordering (ijk) becomes the relation D_{3} + D_{9} + D_{3} = D_{6} + D_{3} + D_{6}. Likewise, (ikj) ↦ D_{6} + D_{1} + D_{11} = D_{3} + D_{5} + D_{1}, (jik) ↦ D_{10} + D_{2} + D_{6} = D_{2} + D_{5} + D_{3}, (jki) ↦ D_{4} + D_{7} + D_{1} = D_{8} + D_{1} + D_{11}, (kij) ↦ D_{2} + D_{7} + D_{4} = D_{10} + D_{2} + D_{8}, and (kji) ↦ D_{8} + D_{4} + D_{8} = D_{4} + D_{12} + D_{4}. After some linear algebra, we find that {D_{1}, D_{2}, D_{6}, D_{8}, D_{9}, D_{11}, D_{12}} form a basis of G_{2}A^{v}(↑), and that the remaining diagrams reduce to the basis as follows: D_{3} = 2D_{6} − D_{9}, D_{4} = 2D_{8} − D_{12}, D_{5} = D_{9} + D_{11} − D_{6}, D_{7} = D_{11} + D_{12} − D_{8}, and D_{10} = D_{11}. In G_{2}A^{sv}(↑) we further have that D_{5} = D_{6}, D_{7} = D_{8}, and D_{9} = D_{10} = D_{11} = D_{12}, and so G_{2}A^{sv}(↑) is 3-dimensional with basis D_{1}, D_{2}, and D_{3} = … = D_{12}. In G_{2}A^{rv}(↑) we further have that D_{5−12} = 0. Thus {D_{1}, D_{2}} is a basis of G_{2}A^{rv}(↑).

There are 3 OC relations to write for G_{2}A^{w}(↑): D_{2} = D_{10}, D_{3} = D_{6}, and D_{4} = D_{8}. Along with the 6T relations, we find that {D_{1}, D_{3} = D_{6} = D_{9}, D_{2} = D_{5} = D_{7} = D_{10} = D_{11}, D_{4} = D_{8} = D_{12}} is a basis of G_{2}A^{w}(↑). Similarly {D_{1}, D_{2} = … = D_{12}} is a basis of the two-dimensional G_{2}A^{sw}(↑). When we mod out by FI, only one diagram remains non-zero in G_{2}A^{rw}(↑) and it is D_{1}.

We leave the determination of the primitives and the spaces with a circle skeleton as an exercise to the reader.

5. Glossary of notation

Greek letters, then Latin, then symbols:

α  maps A^{u} → A^{v} or A^{u} → A^{w}  2.5.5
Δ  cloning, co-product  2.5.1.2
δ  Satoh’s tube map  3.1.1
δ_{A}  a formal D_{A}  3.8
θ  inversion, antipode  2.5.1.1
ι  an inclusion wB_{n} → wB_{n+1}  2.2.3
ι  interpretation map  3.8, 3.8.2
λ  a formal EZ  3.8
ξ_{i}  the generators of F_{n}  2.2.3
Σ  a virtual surface  3.1.1

σ_{i} a crossing between adjacent strands 2.1.1

σ_{ij}  strand i crosses over strand j  2.1.2
ς  the skeleton morphism  2.1.1

(φ^{i}) a basis of 𝔤^{∗} 3.6

ω_{1}  a formal 1-wheel  3.8

A(G) associated graded of G 2.3.2 A_{n}^{−} D_{n}^{v} mod relations 2.3.1 A^{−t} A^{−} allowing trivalent vertices 3.5 A^{−}(↑) D^{v}(↑) mod relations 3.2 A^{−}(◯) A^{−}(↑) for round skeletons 3.3 A^{u} usual chord diagrams 3.9 A(K) the Alexander polynomial 3.7

\overrightarrow{AS}  arrow-AS relations  3.5
a  maps u → v or u → w  2.5.5
a_{ij}  an arrow from i to j  2.3.1
B^{w}  unitrivalent arrow diagrams  3.5
B  the matrix T (exp(−xS) − I)  3.8
b_{ij}^{k}  structure constants of 𝔤^{∗}  3.6

CC the Commutators Commute relation 3.5 D_{n}^{v} arrow diagrams for braids 2.3.1 D^{−t} D^{−} allowing trivalent vertices 3.5 D^{v}(↑) arrow diagrams long knots 3.2

D_{A}  either D_{L} or D_{R}  3.5
D_{L}  left-going isolated arrow  3.5
D_{R}  right-going isolated arrow  3.5
d_{k}  strand deletion  2.5.1.4
d_{i}  the direction of a crossing  3.7
E  the Euler operator  3.8
Ẽ  the normalized Euler operator  3.8
FI  Framing Independence  3.3
F_{n}  the free group  2.2.3

FA_{n} the free associative algebra 2.5.1.5

𝔤  a finite-dimensional Lie algebra  3.6
G_{m}  degree m piece  2.3.1

I  augmentation ideal  2.3.2,
I𝔤  𝔤^{∗} ⋊ 𝔤  3.6

IAM Infinitesimal Alexander

Module 3.8, 3.8.2 IAM^{0} IAM , before relations 3.8.2 \overrightarrow{IHX} arrow-IHX relations 3.5

i_{u}  an inclusion F_{n} → wB_{n+1}  2.2.3
K^{u}  usual knots  3.9
M  the “mixed” move  3.1

OC the Overcrossings Commute relation 2.2 P^{−}(↑) primitives of A^{−}(↑) 3.2 PvB_{n} the group of pure v-braids 2.1.1 PwB_{n} the group of pure w-braids 2.2

R  the relations in IAM  3.8.2
R  Z(�)  2.4
R  the ring ℤ[X, X^{−1}]  3.8.2
R_{1}  the augmentation ideal of R  3.8.2
RI  Rotation number Independence  3.2

R123 Reidemeister moves 3.1 R4 a Reidemeister move for foams/graphs

?? R1^{s} the “spun” R1 move 3.1 S(K) a matrix of signs 3.7 S_{n} the symmetric group 2.1.1 \overrightarrow{ST U} arrow-STU relations 3.5 s_{i} a virtual crossing between adjacent

strands 2.1.1

s_{i}  the sign of a crossing  3.7
sl  self-linking  3.1
T_{𝔤}^{w}  a map A^{w} → U(I𝔤)  3.6
TC  Tails Commute  2.3.1

T (K) the “trapping” matrix 3.7 U universal enveloping algebra 3.6 UC Undercrossings Commute 2.2 u_{k} strand unzips 2.5.1.6 uB_{n} the (usual) braid group 2.1.1 V a finite-type invariant 2.3.1 VR123 virtual Reidemeister moves 3.1 vB_{n} the virtual braid group 2.1.1 W_{m} weight system 2.3.1

w  the map x^{k} ↦ w_{k}  3.7
w_{i}  flip ring #i  2.2.1
w_{k}  the k-wheel  3.5

wB_{n} the group of w-braids 2.2

X  an indeterminate  3.7

X_{n}, X̃_{n} moduli of horizontal rings 2.2.1 x_{i} the generators of F A_{n} 2.5.1.5 (x_{j}) a basis of 𝔤 3.6 Y_{n}, Ỹ_{n} moduli of rings 2.5.4

Z  expansions  throughout
Z^{u}  the Kontsevich integral  3.9

\overrightarrow{4T}  \overrightarrow{4T} relations  2.3.1
6T  6T relations  2.3.1

�, � semi-virtual crossings 2.3.1

⫽  right action  2.2.3
↑  a “long” strand  throughout

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

  E-mail address: drorbn@math.toronto.edu, zsuzsi@math.toronto.edu
  URL: http://www.math.toronto.edu/~drorbn, http://www.math.toronto.edu/zsuzsi