DRAFT

FINITE TYPE INVARIANTS OF W-KNOTTED OBJECTS III: THE DOUBLE TREE CONSTRUCTION

DROR BAR-NATAN AND ZSUZSANNA DANCSO

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

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

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

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

that “there is a formula for V in terms of Φ”, along with an independent topological proof that the said formula works — namely that the equations satisfied by V follow from the equations satisfied by Φ.

Contents

1. Introduction  2
1.1. Topology  2
1.2. Algebra  3
2. The spaces w̃TF and A^{sw} in more detail  5
2.1. The generators of w̃TF  5
2.2. The relations  6
2.3. The operations  7
2.4. The associated graded structure A^{sw}  8
2.5. The homomorphic expansion  11
3. Proof of Theorem ^{\underline{thm:main}}^{\underline{thm:main}} 1.1  13

Date: first edition in future, this edition August 9, 2016. The arXiv:????.???? 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, Kashiwara-Vergne, associators, double tree, free Lie algebras.

This work was partially supported by NSERC grant RGPIN 262178, and by NSF grant no. 0932078 000 while the second author was in residence at the Mathematical Sciences Research Institute in Berkeley, California during Fall 2014. Electronic version and related files at \underline{DoubleTree} [WKO3],

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

DRAFT

3.1. Tree level proof of part (1): Z^{u} determines V^{tree}.  13
3.2. Complete proof of part (1): Z^{u} determines V and C  14

3.3. Proof of part (2): agreement with the ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET] formulas

15

3.4. Proof of part (3): the double tree construction.  19
References  27
Recycling  28
To Do  28

1. Introduction

This is the third in a sequence of papers ^{\underline{Bar-NatanDancso:WKO1, Bar-NatanDancso:WKO2, DoubleTree}} [WKO1, WKO2, WKO3] studying finite type invariants of w-knotted objects, and contains the strongest result: a topological proof for the Kashiwara-Vergne conjecture (more precisely, a topological proof of the ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET] formula for solutions of the Kashiwara-Vergne equations in terms of Drinfel’d associators). Readers familiar with finite type invariants in general should have no trouble reading ^{\underline{Bar-NatanDancso:WKO2}} [WKO2] and this paper without having read ^{\underline{Bar-NatanDancso:WKO1}} [WKO1], however the setup and main results of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2] are used heavily in this paper.

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

K := {uPB ⟶^{cl} sKTG ⟶^{a} w̃TF }.

Let us first briefly elaborate on each of these spaces and maps.

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

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

going to omit the parentheses in the future.  fig:PBexample

DRAFT

− + + + −

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

The space w̃TF is a minor extension of the space wTF studied in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.1 – 4.5], and will be introduced in detail in Section ^{\underline{sec:wTFe}}^{\underline{sec:wTFe}} 2. It can be described as a planar algebra generated by certain features (various kinds of crossings and vertices, as well as “caps”) modulo certain relations (“Reidemeister moves”) and equipped with a number of auxiliary operations beyond planar algebra composition. This Reidemeister theory conjecturally represents knotted tubes in ℝ^{4} with singular “foam vertices”, caps, and attached one-dimensional strings.

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

cl  ^{−} ^{−}^{+}  ^{+}^{+} ^{−}

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

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

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

In Sections 2.2 and 2.3 of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2]we discuss associated graded structures and expansions for general algebraic structures. In our case associated graded structures are always taken with respect to the filtration by powers of the “augmentation ideal”. For the spaces uPB,

DRAFT

sKTG and w̃TF, the associated graded spaces A^{hor}, A^{u} and A^{sw} are the spaces of “horizontal chord diagrams on parenthesized strands”, “chord diagrams on trivalent skeleta”, and “arrow diagrams”, as described in ^{\underline{Bar-Natan:GT1}} [BN1], ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.6], and Section ^{\underline{sec:wTFe}}^{\underline{sec:wTFe}} 2 of this paper, respectively. As a result, the associated graded of K is the structure

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

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

An expansion is a filtration-respecting map from an algebraic structure (where linear combinations of objects of the same kind are allowed) to its associated graded, satisfying a certain non-degeneracy property. In knot theory, expansions are also called universal finite type invariants. A homomorphic expansion is an expansion which behaves well with respect to the operations of the algebraic structure, that is, it intertwines each operation with its induced counterpart on the associated graded structutre. Hence, a homomorphic expansion Z : K → A is a triple of homomorphic expansions Z^{b}, Z^{u}, and Z^{w} for uPB, sKTG and w̃TF, respectively, so that the following diagram commutes:

\xymatrix{uPB \ar[r]^{cl} \ar[d]^{Z^{b}} & sKTG \ar[r]^{a} \ar[d]^{Z^{u}} & w̃TF \ar[d]^{Z^{w}} \\ A^{hor} \ar[r]^{cl} & A^{u} \ar[r]^{α} & A^{w}}  (1)

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

MORE: Clarity and advertisment: state the main theorem from WKO2 here.

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

importantly, there is a formula for V in terms of Φ.

   (2) Said formula is the formula proven in ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET].

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

The key to the proof of the theorem is to show that the generator of w̃TF can be expressed in terms of the generator of uPB and the operations of K. Assuming that Z exists, this yields a formula for V in terms of Φ. The expression of in terms of uses a “Double Tree Construction”, which will be discussed in Section ^{\underline{subsec:DT}}^{\underline{subsec:DT}} 3.4. For now, let us display a picture with no explanation:

DRAFT

cl  ^{a number of these}^{and sKTG ops} in sKTG  in sKTG a, “unzips”, “punctures”, etc. V in w̃TF Φ in uPB

MORE: The plan for this paper is...

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

sec:wTFe

As we mentioned in the introduction, w̃TF is a minor extension of the space wTF studied in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.5]. It can be introduced as a planar algebra or as a circuit algebra; we will do the latter as it is simpler and more concise. Circuit algebras are defined in \underline{Bar-NatanDancso:WKO2} [WKO2, Section 2.4]; in short, they are similar to planar algebras but without the planarity requirement for “connecting strands”. As in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2], each generator and relation of w̃TF has a local topological interpretation. Recall ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Sections 1.2, 3.4, 4.1, 4.5] that wTF diagrams represent certain ribbon knotted tubes with foam vertices in ℝ^{4}, and the circuit algebra wTF is conjecturally a Reidemeister theory for this space (i.e., there is a surjection δ from the circuit algebra wTF to ribbon knotted tubes with foam vertices, and δ is conjectured to be an isomorphism). The space w̃TF extends wTF by adding one-dimensional strands to the picture. Note that one dimensional strands cannot be knotted in ℝ^{4}, however, they can be knotted with the two-dimensional tubes. In figures two-dimensional tubes will be denoted by thick lines and one dimensional strings by thin red lines. With this in mind, we define w̃TF as a circuit algebra defined in terms of generators and relations, and with some extra operations beyond circuit algebra composition. Each generator, relation and operation has a local topological interpretation which provides much of the intuition behind the proofs. However, the corresponding Reidemeister theorem is only conjectural.

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

subsec:wgens

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

The first six generators are as described in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Sections 4.1.1 and 4.5], we only briefly recall their descriptions here. Knotted (more precisely, braided) tubes in ℝ^{4} can equivalently be thought of as movies of flying rings in ℝ^{3}. The two crossings stand for movies where two rings trade places by the ring of the under strand flying through the ring of the over strand. The dotted end represents a tube “capped off” by a disk. Generators 4 and 5 stand for singular “foam vertices”, and will be referred to as the positive and negative vertex, respectively. The positive vertex represents the movie shown on the left: the right ring approaches the left ring from below, flies inside it and merges with it. The negative vertex represents a ring splitting and the inner ring flying out below and to the right. To be completely precise, w̃TF as a circuit algebra has more vertex generators than

DRAFT

OC:  CP:  ↔

A B  A B  A B  A B w  w  w  w =  =
Figure 2. Some w̃TF relations.  fig:wTFeRels

shown above: the vertices appear with all possible orientations of the strands. However, all other versions can be obtained from the ones shown above using “orientation switch” operations (to be discussed in Section ^{\underline{subsec:wops}}^{\underline{subsec:wops}} 2.3). Finally, the generator 6 represents the wen: a Klein bottle cut open, or equivalently, the movie in which a ring flips over. In band notation the wen is shown as a half twist. See ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.5] for a discussion of the wen.

The thin red strands denote one dimensional strings in ℝ^{4}, or “flying points in ℝ^{3}”. The crossings between the two types of strands (generators 6 and 7) represent “points flying through rings”. They are both shown on the right in band notation (see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 3.4] for an explanation of band notation). For example, the bottom left picture means “the point on the right approaches the ring on the left from below, flies through the ring and out to the left above it”. This explains why there are no generators with a thick strand crossing under a thin red strand: a ring cannot fly through a point.

Generator 9 is a trivalent vertex of 1-dimensional strings in ℝ^{4}. Once again, this generator should be shown in all possible strand orientation combinations. Finally, tha last generator is a “mixed vertex”, in other words a one-dimesional string attached to the wall of a 2dimensional tube.

subsec:wrels

2.2. The relations. As a list, the relations for w̃TF are the same as the relations for wTF \underline{Bar-NatanDancso:WKO2} [WKO2, Section 4.5]: \{R1^{s}, R2, R3, R4, OC, CP, FR, W^{2}, CW, TV\}. Recall that R1^{s} is the weak (framed) version of the Reidemeister 1 move; R2 and R3 are the usual Reidemeister moves; R4 allowes moving a strand over or under a vertex. W^{2} states that two consecutive wens can be cancelled, and similarly CW lets a wen on a capped strand disappear (since the cap can slide through the wen). OC stands for Overcorssings Commute, CP for Cap Pullout, FR for Flip Relations, and TV for Twisted Vertex; these four relations are shown in Figure ^{\underline{fig:wTFeRels}}^{\underline{fig:wTFeRels}} 2, for a detailed explanation see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.5].

However, all relations should be interpreted in all possible combinations of strand types (tube or string), for example the lower strand of the R2 relation can either be thick black or thin red, as shown below. Similarly, any of the lower strands of the R3, R4, OC and FR relations may be thin red.

DRAFT

R2:

As in wTF, the relations all have local topological meaning and conjecturally w̃TF is a Reidemeister theory for ribbon knotted tubes in ℝ^{4} with caps, singular foam vertices and attached strings. For example, Reidemeister 2 with a thin red bottom strand is imposed because a point flying in through a ring and then immediately flying back out is isotopic to not having any interaction between the point and ring at all.

subsec:wops

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

The first of these is orientation reversal. For the thin (red) strands, this simply means reversing the direction of the strand. For the thick strands (tubes), orientation switch comes in two versions. Recall from ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 3.4] that in the topological interpretation of wTF, each tube is oriented as a 2-dimensional surface, and also has a distinguished “core”: a line along the tube which is oriented as a 1-dimensional manifold and determines the “direction” or “1-dimensional orientation” of the tube. Both of these are determined by the direction of the strand in the circuit algebra, via Satoh’s tubing map.

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

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

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

e  u_{e}

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

A related operation, disc unzip, is unzip done on a capped strand, pushing the tube off in the direction of the framing (in diagram world, in the direction of the blackboard framing), as before. An example in the line and band notations is shown in Figure ^{\underline{fig:DiscUnzip}}^{\underline{fig:DiscUnzip}} 3; see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.1.3] for details on framings and unzips.

DRAFT

u  u  =
Figure 3. Disc unzip.  fig:DiscUnzip

p p p p p p  e  p_{e}  e p_{e}
Figure 4. Puncture operations.  fig:punctures

Forthermore, one can unzip thin red strands that either end in completely thin red vertices on both sides, or in a thin red vertex at one end and at “infinity” (the tangle boundary) at the other end.

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

For simplicity, we place a restriction on which strands can be punctured, namely at each (fully thick black) vertex puctures are only allowed for one of the three meeting strands, as shown on the left of Figure ^{\underline{fig:punctures}}^{\underline{fig:punctures}} 4. More general puctures could be allowed in a theory with more than one kind of “string to tube” vertex. On the right of the same figure we show what happens when puncturing one of the thick strands of a mixed vertex: the puncture “spreads”. Topologically, this is because the mixed vertex represents a string attached to a tube, so when puncturing e, the entire tube retracts to its core. Finally, a capped tube disappears when punctured, and a wen disappears when the strand which carries it is punctured.

In summary,

w̃TF = CA ⟨ , , , , , ^{w}_{,} , , , | R1^{s}, R2, R3, R4, OC, CP, FR, W^{2}, CW, TV| {S_{e}, A_{e}, \atop u_{e}, d_{e}, p_{e}}⟩

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

w̃TF = CA ⟨ , , , , ^{w}_{,} , , | relations as below | operations as below ⟩ .

Generators 1 and 6 are called single arrows and they are of degree one, while all others are “skeleton features” of degree zero. The relations are almost the same as in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.2.1], which describes the relations for the associated graded of wTF: the skeleton relations W^{2}, TV and CW; \overrightarrow{4T} (the 4-Term relation), TC (Tails Commute), RI (Rotation Invariance), CP (the arrow Cap Pullout), VI (Vertex Invariance), FR (the arrow Flip Relation). For w̃TF there is an additional relation TF (Tails Forbidden on strings). The TC and \overrightarrow{4T} relations are

DRAFT

+ = +  \overrightarrow{4T} =
Figure 5. The TC and \overrightarrow{4T} relations  fig:TCand4T

±  ±  ±  = 0,  and ±  ±  ±  = 0.

Figure 6. The VI relation: the vertices and strands could be of any type, but the same

throughout the relation.  fig:VI

= RI  CP  TF = 0  = 0

FR:  w  = w  but w  = − w .
Figure 7. The RI, CP and FR relations, and the TF relation (which is not really a relation).  fig:RICPTF

shown in Figure ^{\underline{fig:TCand4T}}^{\underline{fig:TCand4T}} 5, note that the 3rd strand in each term of the \overrightarrow{4T} relation is ambiguous: it can be either thick black or thin red, the relation applies in either case. VI is shown in Figure ^{\underline{fig:VI}}^{\underline{fig:VI}} 6: here the ± signs depend on the strand orientations. Note that the type of the vertex and the types of each strand (thick black or thin red) are left ambiguous: the VI relation applies in all cases. Figure ^{\underline{fig:RICPTF}}^{\underline{fig:RICPTF}} 7 shows the other three relations: RI, CP and TF. Note that technically TF is not even a relation: there were no generators with an arrow tail on a thin red strand, so saying that such an element vanishes is superfluous. However, without TF the VI relation would have to be stated for all the sub-cases of 0, 1 or 3 thin red strands, so we prefer this cleaner way of listing the relations even if it is a slight abuse of notation.

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

DRAFT

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

Figure 8. The \overrightarrow{AS}, \overrightarrow{IHX} and the three \overrightarrow{ST U} rerations. Note that in \overrightarrow{ST U}_{1}, the skeleton

strand can be thin red or thick black, and that \overrightarrow{ST U}_{3} “degenerates” to the TC relation.  fig:ASIHXSTU

Figure 9. An example of a “tree”, left, and a “wheel”, right.  fig:TreeAndWheel

To summarise:

w̃TF = CA ⟨ , , , , ^{w}_{,} , , | W^{2}, TV, CW, \overrightarrow{4T}, TC VI, CP, RI, FR, TF| {S_{e}, A_{e}, u_{e}, \atop d_{e}, p_{e}}⟩

As in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Definition 3.7], we define a “w-Jacobi diagram” (or just “arrow diagram”) by also allowing trivalent chord vertices, each of which is equipped with a cyclic orientation, and modulo the \overrightarrow{ST U} relations of Figure ^{\underline{fig:ASIHXSTU}}^{\underline{fig:ASIHXSTU}} 8. Denote the circuit algebra of formal linear combinations of these w-Jacobi diagrams by A^{swt}. Then, as in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Theorem 3.8], we have the following bracket-rise theorem:

Theorem 2.1. The obvious inclusion of diagrams induces a circuit algebra isomorphism A^{sw} ≅ A^{swt}. Furthermore, the \overrightarrow{AS} and \overrightarrow{IHX} relations of Figure ^{\underline{fig:ASIHXSTU}}^{\underline{fig:ASIHXSTU}} 8 hold in A^{swt}.

The proof is identical to the proof of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Theorem 3.8]. In light of this isomorphism, we will drop the extra “t” from the notation and use A^{sw} to denote either of these spaces. As in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2], the primitive elements of A^{sw} are connected diagrams, denoted P^{sw}, and P^{sw} = {trees} ⊕ {wheels} as a vector space. Examples of trees and wheels are shown in Figure ^{\underline{fig:TreeAndWheel}}^{\underline{fig:TreeAndWheel}} 9; for details see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 3.1]. Note that the RI relation can now be rephrased (via \overrightarrow{ST U}_{2}) as “the wheel with a single spoke, or one-wheel, vanishes”.

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

fact:CapIsWheels Fact 2.2. A^{sw}(�), the part of A^{sw} with skeleton a single capped strand, is isomorphic as a vector space to the completed polynomial algebra freely generated by wheels w_{k} with k ≥ 2.

fact:VTwoStrands Fact 2.3. A^{sw}( ) ≅ A^{sw}(↑_{2}), where A^{sw}( ) stands for the space of arrow diagrams whose skeleton is a single vertex (the picture shows a positive vertex but the statement is true for all kinds of vertices with thick black strands), and A^{sw}(↑_{2}) is the space of arrow diagrams on two (thick black) strands.

DRAFT

In addition, we will make use of the following

lem:CapString Lemma 2.4. A^{sw}( ^{...} ) ≅ A^{sw}(↑_{n}), and consequently A^{sw}( ) ≅ A^{sw}(↑_{2}). The left hand side of the first isomorphism is the space of arrow diagrams on n strands whose bottom is capped and a thin red string is attached above the cap. Similarly, the left hand side of the second isomorphism involves a vertex whose two incoming strands are capped with a string attached.

Proof. To prove the first isomorphism, we construct inverse maps between the two spaces. There is an obvious map A^{sw}(↑_{n}) → A^{sw}( ^{...} ), namely, given an arrow diagram on two strands, place it on the capped/stringed strands above the attached string.

In the other direction, given an arrow diagram on capped/stringed strands, first note that in the same vein as Fact ^{\underline{fact:CapIsWheels}}^{\underline{fact:CapIsWheels}} 2.2 we can assume that there are only arrow tails on the capped strand under the attached string. Also, arrow tails die on the thin red strand, hence it has only arrow heads. To produce an arrow diagram on regular strands, first push the arrow tails from the capped strand up and over the string/tube vertex. This is done using the VI relation, but since tails die on the thin red strand they slide past the vertex without a cost. Once the capped side is cleared, continue by pushing the arrow tails up from the thin red string to the strand above the vertex. Note that now the cap kills any arrow heads and hence they also slide past the vertex at no cost. Once the capped and string strands are cleared of any arrow endings, the result can be regarded as an arrow diagram on n regular strands.

It is obvious that the two maps are inverses of each other, so we have proven the first isomorphism. For the second, do the same and combine with Fact ^{\underline{fact:VTwoStrands}}^{\underline{fact:VTwoStrands}} 2.3. □

2.5. The homomorphic expansion. As discussed in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 2.3], an expansion is a map Z^{w} : w̃TF → A^{sw} with the property that the associated graded map gr Z^{w} : A^{sw} → A^{sw} is the identity map id_{A^{sw}}. A homomorphic expansion is an expansion which also intertwines each operation of w̃TF with its arrow diagrammatic counterpart, meaning that the appropriate squares commute. In ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Theorems 4.9 and 4.11] we proved that there exsists a homomorphic expansion for wTF. In fact that homomorphic expansions for wTF (with the minor techical condition that the value of the vertex doesn’t contain isolated arrows) are in one-to-one correspondence with solutions to the Kahiwara-Vergne problem with even Duflo function. We now claim that finding a homomorphic expansion for wTF is equivalent to finding one for w̃TF.

Theorem 2.5. Homomorphic expansions for wTF are in one-to-one correspondence with homomorphic expansions for w̃TF.

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

Proof. For the duration of this proof we introduce the following notation: A^{sw} will denote the associated graded space of wTF and Z^{w} : wTF → A^{sw} a homomorphic expansion. The associated graded space of w̃TF will be denoted by Ã^{sw} and the corresponding homomorphic expansion by Z̃^{w} : w̃TF → Ã^{sw}; see the diagram on the right.

DRAFT

The horizontal maps of the diagram are the obvious inclusions wTF ↪ w̃TF and A^{sw} ↪ Ã^{sw}, namely the maps which send each generator of wTF or A^{w} to the same generator of the corresponding extended space. These maps are injective as none of the relations of w̃TF or Ã^{sw} change the skeletons of the participating diagrams.

e  p_{e}

First assume that Z^{w} is a homomorphic expansion for wTF, we want to show that this extends uniquely to a homomorphic extension of w̃TF. Z^{w} assigns values in A^{sw} to all the generators of wTF, namely the crossings, vertices and cap; hence we only need to define the value of the extension Z̃^{w} on the remaining generators of w̃TF: those which involve thin red strands. Note that if such a homomorphic extension exists, then it is unique: each generator of w̃TF which involves thin red strand can be obtained via puncture operations from generators which do not, see the example in the figure on the right. Since we know the Z̃^{w}-value of all the generators which do not involve thin red strands and Z̃^{w} should be homomorphic with respect to the puncture operation, this determines the value of Z̃^{w} on all generators.

It is straightforward to check that these values satisfy all equations arising from the relations of w̃TF and Z̃^{w} is indeed an expansion. Furthermore, Z̃^{w} is homomorphic by design, using the homomorphicity of Z^{w}.

Conversely, given a homomorphic expansion Z̃^{w} : w̃TF → Ã^{sw}, then the corresponding homomorphic expansion Z^{w} for wTF is the restriction (i.e., the composition of the inclusion of wTF into w̃TF with Z̃^{w}). □

In light of the result above and because this paper focuses on the space w̃TF, we are going to drop the tildes from Ã^{sw} and Z̃^{w}. For the rest of this paper A^{sw} denotes the associated graded space of w̃TF and Z^{w} a homomorphic expansion w̃TF → A^{sw}.

In the proof above we argued that the values of the punctured vertices can be computed from the values of the vertices which already exist in wTF. In fact, more is true: the value of the completely thin red vertex is obviously trivial (thin red strands can only support arrow heads, not tails, hence no arrow diagram can live on a completely thin red skeleton). Moreover, the value of the left punctured vertex turns out to be trivial as well. This fact will be useful later, so we prove it here.

lem:pV Lemma 2.6. For any homomorphic expansion Z^{w}, Z^{w}( ) = 1, that is, the Z^{w}-value of a left punctured vertex is trivial, save for possibly short arrows supported on the right strand^{1}.

Proof. Recall from ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Proof of Theorem 4.9] that the Z^{w}-value of the (positive, not punctured) vertex V can be written as V = e^{b}e^{t}, where b consists of wheels only and t (denoted uD in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2]) consists of trees. Puncturing the left strand of V kills all arrow diagrams with tails on the left. This is almost everything: diagrams that survive are only wheels and short arrows supported entirely on the right strand, and arrows pointing from the right to the left. (We use that if all the tails of a tree are supported on the same strand then the tree is a single arrow, otherwise it is zero by the anti-symmetry of trivalent arrow vertices.) Note that all of the surviving arrow diagrams commute with each other.

Let us then denote the value of the punctured vertex by p_{1}V = e^{p_{1}(b)}e^{p_{1}(t)}. Recall that V must satisfy “unitarity” ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Equation (12)], namely that V · A_{1}A_{2}(V) = 1, where A_{i}

1Which can be ignored without loss of generality, as explained in the last paragraph of the proof.

DRAFT

denotes the adjoint operation performed on strand i. Hence we have p_{1}V · A_{1}A_{2}(p_{1}V) = 1. Since wheels have only tails, A_{1}A_{2}(p_{1}(b)) = p_{1}(b). Each arrow has one head, so A_{1}A_{2}(p_{1}(t)) = −p_{1}(t). Hence, using commutativity, p_{1}V · A_{1}A_{2}(p_{1}V) = e^{2p_{1}(b)} = 1, which implies that p_{1}(b) = 0. As for p_{1}(t), showing that there are no arrows pointing from the right to the left strand is a direct computation in degree 1.

In ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Proof of Theorem 4.9] we showed that short arrows supported on one strand of V don’t affect whether Z^{w} is a homomorphic expansion. In other words, if Z^{w} is a homomorphic expansion and a is a linear combination of such short arrows, then replacing V by e^{a}V gives rise to another homomorphic expansion. Hence we can assume without loss of generality that there are no short arrows, so p_{1}V = 1. □

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

subsec:Part1TreeProof

3.1. Tree level proof of part (1): Z^{u} determines V^{tree}. Let A^{tree} denote the quotient of A^{sw} by all wheels, and let π : A^{sw} → A^{tree} denote the quotient map (cf ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 3.2]). Part (1) of the main theorem states that if a “global” homomorphic expansion Z exists, then it is determined by Z^{u}, in other words, Z^{w} is determined by Z^{u}. Z^{w} on the other hand is given by the values V and C of the positive vertex and the cap ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Sections 4.3 and 4.5], so one only needs to show that V and C are determined by Z^{u}. Proving this “on the tree level” means showing only that π(V) and π(C) are determined by Z^{u}. We prove this partial result first since this contains the main idea without any clouding details. In particular, observe that since C consists entirely of wheels (Fact ^{\underline{fact:CapIsWheels}}^{\underline{fact:CapIsWheels}} 2.2) we have π(C) = 1, so we only need to study π(V).

Let B^{u} denote the “buckle” sKTG, as shown on the right (ignoring the dotted lines). All edges are oriented up, and by the drawing conventions of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.6] all the vertices in the bottom half of the picture are negative and all the ones in the top half are positive. Let B^{w} = a(B^{u}) be the corresponding element of w̃TF. Let β^{u} := Z^{u}(B^{u}), and note that β^{u} can be thought of as a chord diagram on four strands, as VI relations can be used to push all chord endings to the “middle” of the skeleton (between the dotted lines on the picture). We use the shorthand abusive notation β^{u} ∈ A^{u}(↑_{4}), even though four vertical strands don’t form an sKTG. Let β^{w} = α(β^{u}), and note that by the homomorphicity of Z we also have β^{w} = Z^{w}(B^{w}). We are going to perform a series of operations on B^{w} and π(β^{w}) which will allow us to recover π(V) from it.

First, ‘multiply’ B^{w} by a positive vertex at the bottom, as shown in Figure ^{\underline{fig:BuckleToV}}^{\underline{fig:BuckleToV}} 10. This is a circuit algebra operation, and correspondingly π(β^{w}) is “circuit algebra multiplied” by π(V), where V is the value of the vertex. Then unzip the edge marked by u, and puncture the edges marked e and e′ (in that order). Then attach a cap to the thick black end at the bottom. This is a circuit algebra operation; keep in mind that image of the value of the cap is trivial in A^{trees}. Finally, perform a disc unzip on the cap, followed by unzipping the red strand marked by u.

Let us call the resulting w-tangled foam K, as in Figure ^{\underline{fig:BuckleToV}}^{\underline{fig:BuckleToV}} 10. What is Z^{w}(K)? Due to the homomorphicity of Z, it is obtained from β^{w} by performing the series of operations described above. Unfortunately, one of these operations was multiplication by π(V), which is what we are trying to compute. However, notice that the left strand of the added vertex later got punctured, and hence by Lemma ^{\underline{lem:pV}}^{\underline{lem:pV}} 2.6 we only multiplied by 1. (We also multiplied by caps, but as discussed earlier this doesn’t matter for now as π(C) = 1.) Hence, Z^{w}(K) can be

DRAFT

u^{e′}  ^{e}  ^{u}  ^{u}  K
Figure 10. From the “buckle” to the vertex.  fig:BuckleToV

reverse strand 1, connect strands, =  drop cap, rearrange.  = u(C)  S_{1}(V) C  C V  C^{2} C S_{1}(C) S_{1}(u(C))
Figure 11. The Cap Equation and a deduction from it, see explanation below.  fig:CapEqn

computed from β^{w} by performing punctures and unzips, and since β^{w} = α(β^{u}), this means that Z^{w}(K) is determined by Z^{u}.

On the other hand, note that the space of chord diagrams on the skeleton of K is the space A(↑_{2}) by Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.4. Also note that the Z^{w}-value of K in A^{tree}(↑_{2}) is exactly π(V). Hence, π(V) is determined by Z^{u} as needed, and hence Z^{u} determines Z^{w} on the tree level. □

subsec:Part1Proof

3.2. Complete proof of part (1): Z^{u} determines V and C. The main idea of the complete proof is the same as in the tree level proof: we compute Z^{w} from β^{u}. This amounts to computing the Z^{w}-value of the foam K of Figure ^{\underline{fig:BuckleToV}}^{\underline{fig:BuckleToV}} 10 two ways: from β^{w} = α(β^{u}) and from V and C. Then we use the resulting equation to solve for V and C. However, now with one equation and two unknowns, we need some additional information, and this is provided by the Cap Equation ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Equation (13)]. Recall that the Cap Equation is a relationship between V and C which Z^{w} must satisfy in order to be homomorphic with respect to disc unzip. The equation is shown on the left side of Figure ^{\underline{fig:CapEqn}}^{\underline{fig:CapEqn}} 11.

Reverse the first strand of the cap equation and connect the two strands as shown in Figure ^{\underline{fig:CapEqn}}^{\underline{fig:CapEqn}} 11. Note that Fact ^{\underline{fact:CapIsWheels}}^{\underline{fact:CapIsWheels}} 2.2 implies that A^{sw} of a single strand capped on both ends is isomorphic to A^{sw}(�), hence we can ignore the cap at one end of the strand, as in Figure ^{\underline{fig:CapEqn}}^{\underline{fig:CapEqn}} 11. Due to the commutativity of A^{sw}(�), we can rearrange the equation to group all the cap terms on the right, as shown. To see that the new right side of the equation is indeed just C^{2} ∈ A^{sw}(�), we need to note two facts. First, recall from ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 4.5.3] that by CW relation C is even (i.e. consists of even wheels only). Hence, S_{1}(C) = C. Second, note that for any arrow diagram D, S_{1}(u(D)) = 1 when placed on a U-shaped strand as in Figure ^{\underline{fig:CapEqn}}^{\underline{fig:CapEqn}} 11, by defitition of the unzip operation.

Recall from the tree level proof (Section ^{\underline{subsec:Part1TreeProof}}^{\underline{subsec:Part1TreeProof}} 3.1 that Z^{w}(K) = u(C) · p_{1}p_{3}(β^{w}) ∈ A^{sw}(↑_{2}), where p_{1} and p_{3} stand for punctures of the first and third strands of β^{w}. On the other hand, computing Z^{w}(K) directly from V and C, its value is a vertex multiplied by two caps in

DRAFT

=  reverse strand 1 connect and cap  = C  C^{4} u(C) p_{1}p_{3}(β^{w})  C V  S_{1}(p_{1}p_{3}(β^{w}))
Figure 12. Proof of Part 1 of Theorem \underline{thm:main} \underline{thm:main} 1.1.  fig:Part1Proof

A^{sw}(↑_{2}), as shown in Figure ^{\underline{fig:Part1Proof}}^{\underline{fig:Part1Proof}} 12, and these two results are equal by the compatibility of Z^{w} with Z^{u}.

Now reverse strand 1, connect the two strands and cap the top, as shown in Figure ^{\underline{fig:Part1Proof}}^{\underline{fig:Part1Proof}} 12. As before, u(C) cancels on the left hand side, and on the right we get C^{4} since S_{1}(V) = C^{2} in A^{sw}(�). Hence, we have computed C from β^{w}. Now plugging the value of C into the left side equation of Figure ^{\underline{fig:Part1Proof}}^{\underline{fig:Part1Proof}} 12, we find V, hence we have computed V from β^{w}. The claim of Theorem^{\underline{thm:main}}^{\underline{thm:main}} 1.1 was that we can compute V from associators: indeed, it is easy to write β^{w} in terms of associators, as we shall see shortly in Section ^{\underline{subsec:AETFormula}}^{\underline{subsec:AETFormula}} 3.3. □

subsec:AETFormula

3.3. Proof of part (2): agreement with the ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET] formulas. First let us note that the \underline{AlekseevEnriquezTorossian:ExplicitSolutions} [AET] formulas only concern the tree-level part π(V). To be more precise, let V = e^{b}e^{u(D)} as in Section 4.4 of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2], where b is a “power series” of wheels in the sense of Fact ^{\underline{fact:CapIsWheels}}^{\underline{fact:CapIsWheels}} 2.2, and u(D) is a primitive element in A^{tree}. Then F = e^{D} ∈ T Aut_{2} is a solution to the Kashiwara-Vergne problem in the sense of ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET]. For details see^{2} Section 4 of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2]. Thus, to

match our formulas to those of ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET], we only need to worry about π(V). In Section ^{\underline{subsec:Part1TreeProof}}^{\underline{subsec:Part1TreeProof}} 3.1,

we constructed π(V) from the “buckle” Z^{u}(B^{u}) = β^{u}. Namely, π(V) is given by παβ^{u} with strands 1 and 3 punctured and 2 and 4 capped, interpreted as a diagram on two strands via

the isomorphism of Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.4.

We start with computing β^{u}. B^{u} is the closure of the parenthesized braid B^{b} shown in Figure ^{\underline{fig:BuckleBraid}}^{\underline{fig:BuckleBraid}} 13, therefore it is enough to compute β^{b} := Z^{b}(B^{b}). The result can be read off the

picture in Figure ^{\underline{fig:BuckleBraid}}^{\underline{fig:BuckleBraid}} 13:

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

To interpret this formula, recall that the associator Φ is an element of A^{hor}(↑_{3}), and the subscripts show which strands diagrams are placed on (this is often done in the literature using superscripts instead). For example, the notation Φ_{(13)24}^{−1} means doubling the first strand of Φ^{−1} and placing the resulting chord endings on starnds 1 and 3, as well as placing the chord endings from the other two strands of Φ^{−1} on strands 2 and 4. Also recall that R = e^{c/2}, where c is a single horizontal chord between two strands (and in this case R_{32} means that this chord runs between strands 3 and 2). Note that in figures we number the strands at the top and read multiplication bottom to top.

As β^{u} is the tree closure of β^{b}, it is given by the same formula interpreted as an element of A^{u}(↑_{4}). In turn, β^{w} = α(β^{u}), and to compute π(V), we puncture strands 1 and 3 of β^{w}, and cap strands 2 and 4. Note that p_{3}α(R_{32}) = e^{a_{23}/2}, where a_{23} is a single arrow pointing from strand 2 to strand 3.

2There are some notational differences between \underline{AlekseevTorossian:KashiwaraVergne} [AT] and \underline{AlekseevEnriquezTorossian:ExplicitSolutions} [AET], hence we don’t switch strands here. The value V and associator Φ are the same throughout this series of papers, in particular they agree with \underline{Bar-NatanDancso:WKO2} [WKO2] and \underline{Bar-Natan:WKO4} [WKO4]. For computatinal implementation and verifications see \underline{Bar-Natan:WKO4} [WKO4].

DRAFT

1 ^{2}  ^{3} ^{4}  Φ_{(13)24}^{−1} Φ_{132} R_{32} Φ_{123}^{−1} Φ_{(12)34}
Figure 13. Computing β^{b}.  fig:BuckleBraid

Recall that Φ_{123} is a horizontal chord associator which can be expressed as a power series in non-commuting variables c_{12} and c_{23} (i.e., chords between strands 1—2 and 2—3, respectively). The image of Φ in the quotient where c_{12} and c_{23} commute is 1. Hence, p_{1}p_{3}(αΦ_{123}^{−1}) = 1, as the punctured strands can only support heads, and tails on the middle strand commute by TC. Similarly, p_{1}p_{3}(αΦ_{132}) = 1 because the punctures kill the entire “left side” of the associator (that is, p_{1}p_{3}α(c_{13}) = 0).

Thus p_{1}p_{3}(β^{w}) can be expressed as α(Φ_{(13)24}^{−1})e^{a_{23}/2}α(Φ_{(12)34}). Since strands 1 and 3 are punctured, no arrows can be supported between these strands, hence p_{1}p_{3}α(Φ_{(12)34}) = p_{1}p_{3}α(Φ_{234}).

Observe that the punctured strand 3 can only support arrow heads. Expressing Φ as a power series in two variables, Φ_{234} = Φ(c_{23}, c_{34}), and p_{1}p_{3}α(Φ(c_{23}, c_{34})) = Φ(a_{23}, a_{43}), where a_{23} is an arrow pointing from strand 2 to strand 3, and a_{43} is an arrow pointing from strand 4 to strand 3. Similarly, Φ_{(13)24}^{−1} = Φ^{−1}(c_{(13)2}, c_{24}), where c_{(13)2} = c_{12} + c_{32}. In turn, by a property of associators, Φ^{−1}(c_{(13)2}, c_{24}) = Φ^{−1}(c_{(13)2}, −c_{(13)2} − c_{(13)4}), and so p_{1}p_{3}αΦ_{(13)24}^{−1} = Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}).

To summarise,

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

Now we need to interpret this result as an element of A^{w}(↑_{2}) via Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.4. The first strand of A^{w}(↑_{2}) joins strands 1 and 2 in a vertex, and the second strand of A^{w}(↑_{2}) joins strands 3 and 4. Let us call the two strands of A^{w}(↑_{2}) strand I and strand II to avoid confusion. Recall from the proof of Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.4 that one first slides arrow tails “up” through the vertices from the capped strands 2 and 4 (according to the forulas above there are no arrow heads on the capped strands), then one slides all the heads up from the punctured strands 1 and 3. Hence, one obtains an element of A^{w}(↑_{2}) in which all arrow heads are below all tails on both strands. The result is shown in Figure ^{\underline{fig:ValueV}}^{\underline{fig:ValueV}} 14.

Next, we need to find the corresponding element F ∈ TAut_{2}, and check that it agrees with the formula of ^{\underline{AlekseevEnriquezTorossian:ExplicitSolutions}} [AET, Theorem 4]. According to this formula^{3},

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

meaning that if x and y are the generators of the degree completed free Lie algebra 𝔩𝔦𝔢_{2}, then F conjugates x by Φ^{−1}(x, −x − y), and conjugates y by e^{−(x+y)/2}Φ(−x − y, y)e^{y/2}.

3There are sign differences from \underline{AlekseevEnriquezTorossian:ExplicitSolutions} [AET] due to notational misalignment, for example our Φ is \underline{AlekseevEnriquezTorossian:ExplicitSolutions} [AET]’s Φ^{−1}.

Our notation is consistent with all other papers in this series and the formulas are computationally verified in \underline{Bar-Natan:WKO4} [WKO4].

DRAFT

Φ_{1}^{−1} Φ_{2}^{−1} Φ_{2}  Φ_{4} Φ_{4}^{−1} Φ_{3} Φ_{3}^{−1} e^{a}  ^{TAILS}^{HEADS} I  II

Figure 14. The value of π(V) in terms of associators. Here Φ_{1}^{−1} stands for all the arrow heads on strand 1 in Φ^{−1}(a_{2(13)}, −a_{2(13)} − a_{4(13)}). Similarly Φ_{2}^{−1} denotes the arrow tails on

strand 2 in the same formula, Φ_{2} stands for the arrow tails on strand 2 of Φ(a_{23}, a_{43}), etc.  fig:ValueV
\xymatrix{U(𝔩𝔦𝔢_{2}^{2}) \ar[dd]^{L} & U(𝔩𝔦𝔢_{2}^{2})_{exp} \ar@{^{(}->}[l] \ar[dd]^{L} \ar[dr]^{φ} \\  &  & _{TAut_{2}} \\ A^{w}(↑_{2}) & A^{w}(↑_{2})_{exp} \ar@{^{(}->}[l] \ar@{-->}[ur]^{ξ}}

Section 3.2 of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2] described the relationship between the primitives of A^{w}(↑_{n}) and 𝔱𝔡𝔢𝔯_{n} in detail, and familiarity with it those results will be helful, but we summarize the relevant facts about the relationship between A^{w}(↑_{2}) and TAut_{2} below.

The map φ is the (^{\underline{eq:AET}}^{\underline{eq:AET}} 2) interpretation map: an element (e^{λ_{1}}, e^{λ_{2}}) ∈ U(𝔩𝔦𝔢_{2}^{2})_{exp} is mapped to the automorphism of 𝔩𝔦𝔢_{2} which sends the generator x of 𝔩𝔦𝔢_{2} to e^{−λ_{1}}xe^{λ_{1}}, and the generator y to e^{−λ_{2}}ye^{λ_{2}}. Beware: this is not a group homomorphism! In other words, composition in TAut_{2} is not given by piecewise multiplication of the conjugators.

The map L is defined as follows. A primitive element (λ_{1}, λ_{2}) of 𝔩𝔦𝔢_{2}^{2} can be represented by a linear combination “floating trees” with symmetrized ends coloured by the symbols x and y. A “floating tree” is an oriented uni-trivalent graph with two-in-one-out vertices equipped with a cyclic orientation. The leaves (tails) are coloured by the symbols x and y, and hence modulo \overrightarrow{AS} and \overrightarrow{IHX} each tree corresponds to an element in 𝔩𝔦𝔢_{2}. The heads are also coloured by x or y, so there are two kinds of such trees, that is, two factors of 𝔩𝔦𝔢_{2}. In other words, λ_{1} represents a (possibly infinite) linear combination of trees whose head is coloured by x and λ_{2} represents a linear combination of trees whose head is coloured by y. (Cf. primitive elements of B_{2}^{w} as in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2][Theorem 3.16] and the discussion following it.)

Floating trees can be mapped into A^{w}(↑_{2}) by “attaching its ends” to the appropriate strands, namely, the ends labelled x to first strand and the ends labelled y to the second strand. There is a choice involved in where to attach the head of the tree in relation to the tails on the same strand. In ^{\underline{Bar-NatanDancso:WKO2}} [WKO2][Proof of Proposition 3.19] we defined a map l which attaches the head of each tree below (lower than) all of its tails; l is a Lie algebra map from 𝔩𝔦𝔢_{2}^{2} to the primitives of A^{w}(↑_{2}).

The map L is an extension of l to U(𝔩𝔦𝔢_{2}^{2}): an element of U(𝔩𝔦𝔢_{2}^{2}) is a sum of products (disjoint unions) of coloured trees and L attaches the ends to the appropriate strands by

DRAFT

π(V)^{−1} π(V)  x
Figure 15. The action of π(V) on the generator x of 𝔩𝔦𝔢_{2}.  fig:ActOnx

Φ_{(24)13}^{−1} Φ_{241} R_{14}^{−1} Φ_{214}^{−1} Φ_{(21)43} R_{(13)(24)} R_{12}^{−1}R_{34}^{−1}^{1}^{2}  ^{3}^{4}
Figure 16. A different expression of β^{b}.  fig:BuckleBraid2

attaching all of the arrow heads in a product below all of the arrow tails. Beware: L ≠ exp(l), and L is not a homomorphism of algebras or groups.

However, the maps L and φ fail to be group homomorphisms in exactly the same way, in the following sense: there is a group homomorphism ξ : A^{w}(↑_{2})_{exp} → TAut_{2} which makes the triangle commute. The map ξ is the exponential of the action of the primitives of A^{w}(↑_{2}) on 𝔩𝔦𝔢_{2} described in the “Conceptual argument” for ^{\underline{Bar-NatanDancso:WKO2}} [WKO2][Proposition 3.19]. Namely, an element D ∈ A^{w}(↑_{2})_{exp} acts on 𝔩𝔦𝔢_{2} the following way. Add an extra strand. Reresent an element v ∈ 𝔩𝔦𝔢_{2} by a tree whose head is on the extra strand (more precisely, a linear combination of such trees). Its conjugae D^{−1}vD is once again a linear combination of such trees, defined to be the output of the action. The reader can check that this is indeed a hoomorphism and it makes the triangle commute.

Now we are ready to compute how π(V) ∈ A^{w}(↑_{2}) acts on the generator x of 𝔩𝔦𝔢_{2} and match this to the formula ^{\underline{eq:AET}}^{\underline{eq:AET}} 2. Recall the value of π(V) shown in Figure ^{\underline{fig:ValueV}}^{\underline{fig:ValueV}} 14. The generator x is represented by an arrow from the first strand to the added third strand, and the result of the action is π(V)^{−1}xπ(V), as shown in Figure ^{\underline{fig:ActOnx}}^{\underline{fig:ActOnx}} 15. To compute this, one commutes the tail of x up to the top of the strand across π(V) using \overrightarrow{ST U} relations. At the end of this process, π(V) and π(V)^{−1} cancel, and the result of the action remains. Observe that due to the T C relation, only the heads on strand I act on x, in other words the action is by Φ_{124}^{−1} = Φ^{−1}(a_{21}, −a_{21} − a_{41}), arranged on strands I and II as shown in Figure ^{\underline{fig:ValueV}}^{\underline{fig:ValueV}} 14. An Lpreimage of this element is exactly Φ^{−1}(x, −x − y) in the first component, as in Formula (^{\underline{eq:AET}}^{\underline{eq:AET}} 2).

As for the action of F on y, using the expression of Figure ^{\underline{fig:BuckleBraid}}^{\underline{fig:BuckleBraid}} 13 results in a formula different from but equivalent to (^{\underline{eq:AET}}^{\underline{eq:AET}} 2). To obtain (^{\underline{eq:AET}}^{\underline{eq:AET}} 2), one needs to use a different expression of the “buckle braid” β^{b}, as shown in Figure ^{\underline{fig:BuckleBraid2}}^{\underline{fig:BuckleBraid2}} 16.

DRAFT

Simplifying the formula as before, we have that after the cap and puncture operations R_{(13)(24)} = e^{a_{(24)(13)}/2}, Φ_{(24)13}^{−1} = 1, Φ_{241} = Φ(−a_{41} − a_{21}, a_{41}), R_{14}^{−1} = e^{−a_{41}/2}, Φ_{214}^{−1} = Φ^{−1}(a_{21}, a_{41}), Φ_{(21)43} = Φ(−a_{43} − a_{23}, a_{43}), R_{12}^{−1} = e^{−a_{21}/2}, and R_{34}^{−1} = e^{−a_{43}/2}.

Using Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.4 slide arrow endings up to strands I and II, first from strands 2 and 4, then from strands 1 and 3. Since we are computing the action on y, the only part of the result which matters is arrow heads on strand II. These all come from strand 3, namely, read from bottom to top they are e^{(a_{23}+a_{43})/2}, followed by Φ(−a_{43} − a_{23}, a_{43}), followed by e^{−a_{43}/2}. Taking the L-preimage we obtain e^{(x+y)/2}Φ(−x − y, y)e^{−y/2}, as stated in Formula ^{\underline{eq:AET}}^{\underline{eq:AET}} 2.

subsec:DT

3.4. Proof of part (3): the double tree construction. What remains is to prove that the values of V and C above indeed give rise to a homomorphic expansion of w̃TF. Unfortunately, checking directly that they satisfy all necessary equations is very difficult.

Note that most of the equations arise from relations that assert an equality between different planar algebra compositions of generators. For example, the crucial R4 relation is of this kind. Hence, life would be much easier if Z^{u} were to be a planar algebra map. This unfortunately makes no sense, as sKTG is not a planar algebra but a different, more complicated algebraic structure. The reader might ask, why work with a space as inconvenient as sKTG instead of, say, the planar algebra of ordinary (or usual) trivalent tangles? Then both the definition of V from Z^{u} and the proof that the resulting Z^{w} is a homomorphic expansion would have been much simpler. The answer is that the existence of a homomorphic expansion is a highly non-trivial property that is not guaranteed to be the case, and in particular ordinary trivalent tangles do not have one. Even without trivalent vertices, ordinary tangles, or u-tangles, do not have have one. Only parenthesized tangles (a.k.a. q-tangles) do, and in fact these are almost equivalent to sKTG.

Nonetheless, the dreams above are not entirely futile, as they lead to the following strategy: we map usual trivalent tangles into sKTG via a double tree construction, and use this to define Z^{w} for the a-image of all usual trivalent tangles. Then we’ll use the planar algebra structure to prove that this Z^{w} is a homomorphic expansion, and finally show that this Z^{w} is in fact the same as the one arising from part (1).

We start by defining usual trivalent tangles, to be denoted uT T:

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

Here PA stands for planar algebra, which in our context is the same as a circuit algebra, except the wiring diagrams defining the operations must be planar. This is a slightly simpler notion than the standard use of the term in the literature, in particular we do not use checkerboard shadings. The elements of uT T can be seen as usual (i.e., classical) trivalent tangles with signed vertices and no virtual crossings, modulo all of the relations which make sense in this context.

We define a double tree map � : uT T → sKTG, in Figure ^{\underline{fig:dt}}^{\underline{fig:dt}} 17. The � map depends on two choices of binary trees: in Figure ^{\underline{fig:dt}}^{\underline{fig:dt}} 17 we chose a particular example. It is important that, regardless of this choice, the “left side” tree always crosses over the “right side” tree. We will soon demonstrate that in fact the choice of trees does not matter: we will compose � with Z^{u} followd by a number of further operations, as shown in Equation (^{\underline{eq:dt}}^{\underline{eq:dt}} 3), and we’ll show that this composition is well-defined and independent of the choices made. It will later be

DRAFT

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

Figure 17. The double tree map: connect the ends of T by two binary trees (hence “double tree”), as shown. Note that the tree on the left always crosses over the tree on the right,

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

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

Figure 18. The effect of punctures, caps and unzips. The notation here is slightly abusive: these operations are applied after Z^{u} and α, in A^{w}. Nonetheless, we pretend here that they are applied to a(�(T)). On one hand the effect on the skeleton, which is our main focus

here, is the same; on the other hand the lack of arrow diagrams makes the pictures simpler.  fig:pCu

important that we can choose any tree for �, as opposed to fixing a particular choice, such as the one in Figure ^{\underline{fig:dt}}^{\underline{fig:dt}} 17.

T ∈ uT T ⟶^{�} sKTG ⟶^{Z^{u}} A^{u}(�(T)) ⟶^{α} A^{w}(�(T)) ⟶^{C,p,u} A^{w}(Ť) ≅ A^{w}(T).  (3) eq:dt

Here let T denote an arbitrary tangle in uT T. The double tree construction maps T into sKTG, and by applying Z^{u} one obtains a value in A^{u}, namely a chord diagram on the skeleton of �(T). By a slight abuse of notation we denote this space by A^{u}(�(T)). Now α maps into arrow diagrams on the skeleton of �(T), that is, A^{w}(�(T)). We then apply a number of operations in A^{w}: a cap attachment, punctures and unzips (as shown in Figure ^{\underline{fig:pCu}}^{\underline{fig:pCu}} 18 and explained in the next paragraph), which reverts the skeleton back to a slightly modified version of the skeleton of T, denoted by Ť. Finally, we claim that A^{w}(Ť) ≅ A^{w}(T) via the isomorphism of Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.4, and hence we obtain a value in A^{w}(T), as needed.

The puncture, cap and unzip operations are done in the following order: first attach a cap to the end of the right vertical strand (this is a circuit algebra operation). Then puncture the strand marked with “1” on the left of Figure ^{\underline{fig:pCu}}^{\underline{fig:pCu}} 18. Then puncture strands 2, 3, and 4 in that order, and recall that these punctures also spread to the connecting diagonal strands, as in Figure ^{\underline{fig:punctures}}^{\underline{fig:punctures}} 4. Finally, perform disc unzips on the right vertical strand and unzip the punctured thin red string until there are no more tree vertices other than the mixed ones on top, as

DRAFT

shown in Figure ^{\underline{fig:pCu}}^{\underline{fig:pCu}} 18. Note that since the punctured tree had originally always crossed over the capped tree, the crossings all become virtual after the punctures, hence the equality in

Figure ^{\underline{fig:pCu}}^{\underline{fig:pCu}} 18.

Let us denote the composition of the maps and operations shown in Equation (^{\underline{eq:dt}}^{\underline{eq:dt}} 3) by ξ, that is, ξ = u ∘ p ∘ C ∘ α ∘ Z^{u} ∘ �, and we can say that ξ(T) ∈ A^{w}(T) via the isomorphism of Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.4. Our next task is to show that ξ(T) is well-defined, that is, it doesn’t depend on the choice of binary trees.

lem:TreeChange Lemma 3.1. The choice of binary trees in the double tree construction does not affect ξ(T).

Proof. Any binary tree can be changed into any other binary tree via a sequence of “I to H” moves, as shown on the right. Hence, it is enough to analyze how an I to H move on one of the trees affects the value of Z^{u}(�(T)), and prove that the difference created by an I to H move does not “survive” through the cap and puncture operations.

2 ^{1}  u^{2}

Suppose τ_{1} and τ_{2} are two trees which differ by a single I to H move, and let �_{τ_{1}} and �_{τ_{2}} denote the two resulting double-tree maps, assuming that the other tree is unchanged. The I to H move can be realised by inserting an associator, followed by unzipping first the edge marked ‘1’ on the right, then the edge marked ‘2’. See ^{\underline{Bar-NatanDancso:WKO2}} [WKO2][Section 4.6] for a detailed description of tangle insertion. Hence, by the homomorphicity of Z^{u}, the values Z^{u}(�_{τ_{2}}(T)) and Z^{u}(�_{τ_{1}}(T)), only differ in an inserted horizontal chord associator Φ on the three strands involved. In sloppy notation, Z^{u}(�_{τ_{2}}(T)) = Z^{u}(�_{τ_{1}}(T)) ∗ Φ. If the I to H move was done on the left side tree, then all the strands involved are later punctured, killing any arrow diagram that lived on them by the TF relation. As a result, the only surviving part of Φ is its constant term, 1, and the resulting values of ξ are equal.

T τ_{2} τ_{1} \overline{τ_{1}}

If the I to H move happened on the right side tree, then the all participating strands are capped and disk unzipped. Arrow heads adjacent to a cap are zero by the CP relation, but there may be other arrow endings separating α(Φ) from the caps, preventing it from being eliminated. To show that this is not the case, we use the trick shown on the right: �_{τ_{2}}(T) can be achieved by attaching τ_{1} under T, then τ_{1} upside down (denoted \overline{τ_{1}}), then τ_{2}. Unzip the edge connecting τ_{1} and \overline{τ_{1}} several times until all vertices of τ_{1} are gone, the result is just straight lines connecting T and τ_{2}, that is, �_{τ_{2}}(T).

τ_{2} \overline{τ_{1}}

Now recall that τ_{2} is marely τ_{1} with an associator inserted (and unzipped). In the language of ^{\underline{Bar-NatanDancso:WKO2}} [WKO2, Section 5.2], \overline{τ_{1}}τ_{2} consists of some “nested bubbles”, with a single associator inserted, as shown on the right. Since Z^{u} of the bubble is trivial and Z^{u} is compatible with insertions, the only chords in Z^{u}(\overline{τ_{1}}τ_{2}) are the ones in Z^{u}(Φ). Hence, after the cap attachment and disk unzips, α(Φ) is directly adjacent to the caps, and since Φ is a horizontal chord associator, all positive degree parts of α(Φ) become zero due to the CP relation. This concludes the proof. □

Observe that there is an action of ℤ/nℤ on elements of uT T with n ends: namely, the cyclic permutations of the ends.

lem:CyclicPerm Lemma 3.2. The map ξ is invariant under cyclic permutation of the ends of T.

DRAFT

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

T^{unzip}  unzips  T �(σT) with inserts:  �(T)
Figure 20. The difference between �(T) and �(σT), understood via insertions.  fig:welldef2

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

The rightmost picture of Figure ^{\underline{fig:welldef}}^{\underline{fig:welldef}} 19 is equivalent as sKTGs to �(σT). It differs from �(T) in three ways: the binary trees connecting the ends of T are different; two tree branches are connected to the trunk “on the wrong side” (marked by ∗ in Figure ^{\underline{fig:welldef}}^{\underline{fig:welldef}} 19), and one tree branch has a “kink” in it. As before, we need to analyse how Z^{u}(�(σT)) differs from Z^{u}(�(T)), and show that the difference doesn’t survive through the puncture, cap and unzip operations.

To achieve this, we transform �(σT) into �(T) using tangle insertions. First, cancel the kink by inserting the opposite kink on the same strand, as shown in Figure ^{\underline{fig:welldef2}}^{\underline{fig:welldef2}} 20 in blue (grey in black and white print). As Z^{u} is compatible with insertion, the Z^{u} values will differ by the value of a kink, a chord diagram on the one strand involved. However, note that later in the process this strand is punctured, and recall that no arrow diagram can live on punctured strands only (by the TF relation). Hence, after applying α and puncturing, the value of the kink is 1.

Similarly, switching the side that the tree branches are attached on amounts to inserting twists and unzipping the connecting edges, as also shown in Figure ^{\underline{fig:welldef2}}^{\underline{fig:welldef2}} 20. Each of these operations changes the value of Z^{u} by inserting the value of a twist: e^{c/2}, where c denotes a single chord between the appropriate strands. Applying α maps this to e^{(a_{L}+a_{R})/2}, where a_{L} and a_{R} denote horizontal left and right arrows, respectively. However, once the left side

DRAFT

T T  Z^{u}  Z^{w}  Z^{w}  Z^{w}  Z^{w} T  Z^{w}(T) T  T T  T ↦^{�}  ↦^{·C} ↦^{α}  ↦^{p}  ↦^{u}  ↦^{≅} T T  ↦^{≅} ↦^{Z^{w}}  Z^{w}(T) = T  T  Ť T ↦^{a}  ↦^{u} ↦^{·C}  ↦^{p} C cancels after puncture, see proof. ξ  C C C  C C C C

Figure 21. The compatibility of Z^{w} with Z^{u} and various operations. The chords and arrows

are meant as schematic indications of chord and arrow diagrams, not as actual values.  fig:BigCompat

tree is punctured, the value of the twist on it becomes 1, as before. The strand under the bottom right twist is then capped and uzipped, hence that value also becomes 1 due to the CP relation.

Now observe that the right side picture of Figure ^{\underline{fig:welldef2}}^{\underline{fig:welldef2}} 20 only differs from �(T) in the choices of binary trees, which do not change the value of ξ by Lemma ^{\underline{lem:TreeChange}}^{\underline{lem:TreeChange}} 3.1. □

lem:Compatibility Lemma 3.3. The value of ξ depends only on Z^{u}. Furthermore, if there exists a homomorphic expansion Z^{w} for w̃TF, then for a tangle T ∈ a(uT T), the value Z^{w}(T) is equal to ξ(T) multiplied by C^{−1} at each tangle end of T, where C is the Z^{w}-value of the cap.

Proof. If there exists a homomorphic expansion Z^{w} wich is compatible with Z^{u}, then we can use, as in Figure ^{\underline{fig:BigCompat}}^{\underline{fig:BigCompat}} 21, the homomorphicity of Z^{w} and its compatibility with Z^{u} to show that ξ(T) = Z^{w}(Ť). To explain the Z^{w} arrow of the top row of the figure, note that as Z^{w} is a circuit algebra homomorphism, Z^{w}(Ť) can be obtained from Z^{w}(T) by multiplying with a left-punctured right-capped vertex at each tangle end. By Lemma ^{\underline{lem:pV}}^{\underline{lem:pV}} 2.6 the value of the left-punctured vertex is 1, so we are simply attaching the value C of the cap at each end, as shown. Also recall that A^{w}(T) ≅ A^{w}(Ť) by Lemma ^{\underline{lem:CapString}}^{\underline{lem:CapString}} 2.4, explaining the last horizontal arrow of each row. This proves the second statement.

T  T = In imα C  C

The proof of the first statement is the claim in Figure ^{\underline{fig:BigCompat}}^{\underline{fig:BigCompat}} 21 stating that the value of the cap introduced in carrying out ξ on T cancels. This is a consequence of the “tail-invariance” of arrow diagrams in the image of α (see ^{\underline{Bar-NatanDancso:WKO2}} [WKO2][Remark 3.11 and early Section 3.3]. In short, as long as a tangle is in the image of a, one can slide a strand freely under it. The arrow diagram implication of this fact is the tail invariance of arrow diagrams in the image of α, which in our situation means that the value C in the diagram can be moved to the other side, as shown on the right. Consequently, C cancels in the next step when this strand is punctured. □

Strategic preview: To summarize, if Z^{w} exists, then we can compute C by evaluating ξ on a single un-knotted strand (denote this by ↑): since Z^{w}(↑) = 1 for any homomorphic

DRAFT

↦^{�}  ^{+}^{−}^{+}^{−}  ↦^{Z^{u}}  ^{Φ^{−1}}^{Φ}  ^{n}^{b}^{Φ}^{Φ^{−1}}^{R^{−1}}
Figure 22. Double tree composed with Z^{u}, applied to a single strand.  fig:dtstrand

expansion, we have ξ(↑) = C^{2}, determining the value of C. To then compute the value V of the vertex, observe that the vertex is an element of uT T, so evaluate ξ on it, then multiply by C^{−1} at each end. We then have to show that the resulting Z^{w} is indeed a homomorphic expansion of w̃TF, and in order to do this we’ll show that it is a planar algebra homomorphism. The compatibility of Z^{w} with Z^{u} is automatic by design.

We start by computing ξ(↑) and C from it. Recall Fact ^{\underline{fact:CapIsWheels}}^{\underline{fact:CapIsWheels}} 2.2, which implies that C involves wheels only, so ξ(↑) must consist only of wheels if it is to equal C^{2}. This will also serve as a first reality check.

lem:xiofstrand Lemma 3.4. For the un-knotted strand, ξ(↑) = α(ν^{1/2}). Here ν ∈ A^{u}(↑) denotes the Kontsevich integral of the un-knot, which indeed involves wheels only^{4}.

Proof. We apply � to ↑, as shown in Figure ^{\underline{fig:dtstrand}}^{\underline{fig:dtstrand}} 22. Our first goal is to comupte Z^{u}(�(↑)). In \underline{Bar-NatanDancso:WKO2} [WKO2][Section 5.2] we give an algorithm for writing any sKTG as a “porduct” of generators, and hence expressing its Z^{u} value in terms of the Z^{u}-values of the generators. To run �(↑) through this algorithm, one end of the tangle needs to be “on top”, so we curve up the strand on the right (the choice of side doesn’t change the result).

The results of applying � and Z^{u} are shown in Figure ^{\underline{fig:dtstrand}}^{\underline{fig:dtstrand}} 22. The value of Z^{u} is expressed in terms of Φ, the value of the associator graph and a horizontal-chord Drinfel’d associator, the value of the twist R = e^{c/2} where c is a single chord, and the values n and b of the noose and balloon graphs, respectively. See ^{\underline{Bar-NatanDancso:WKO2}} [WKO2][Section 4.6] for details.

In order to compute ξ(↑), Z^{u} is followed by α, a cap attachment, punctures and unzips. As explained in ^{\underline{Bar-NatanDancso:WKO2}} [WKO2][Section 4.6], there is a one-parameter uncertainty in the exact values of n and b, but we do know that and α(b) = e^{a/2}α(ν)^{1/2}, and α(n) = e^{a/2}α(ν)^{1/2}. Note that the exponential part of n cancels once the cap is attached by the CP relation. Let us “push” most of the arrow diagrams to the middle four strands using the VI relation. The result after α, cap, puncture and unzip is shown in Figure ^{\underline{fig:dtstrand2}}^{\underline{fig:dtstrand2}} 23 and explained below.

Note that α maps each chord to the sum of its two possible orientations. However, when one of the strands supporting it is punctured, the only surviving orientation (due to the CP) relation is the arrow which has its head on the punctured strand. Hence, for example, α(R^{−1}) = (e^{−c/2})_{23} simplifies to (e^{−a/2})_{32}. Recall that the subscripts denote the strands on which diagrams live, as in Figure ^{\underline{fig:BuckleBraid}}^{\underline{fig:BuckleBraid}} 13, and note that the figure is over-simplified, as exponentials and associators are denoted by single arrows. As for associators, recall that

4The value of ν was conjectured in \underline{Bar-NatanGaroufalidisRozanskyThurston:WheelsWheeling} [BGRT] and proven in \underline{Bar-NatanLeThurston:TwoApplications} [BLT].

DRAFT

1  ^{3} ^{4}^{2}  ^{α(ν^{1/2})_{(34)}}^{α(Φ)_{12(34)}}^{α(Φ^{−1})_{234}}^{(e^{−a/2})_{32} from α(R^{−1})}^{α(Φ^{−1})_{134}} 5 ^{e^{−a/2}}  ^{α(Φ)_{324}}^{+}^{α(ν^{1/2})}
Figure 23. The result after applying α, cap, puncture and unzips.  fig:dtstrand2

Φ ∈ A^{h}or(↑_{3}) can be written as a power series in any two of the three generators of A^{hor}(↑_{3}), namely c_{12}, c_{23} and c_{13}. For each associator above, we chose the presentation in which α(Φ) is of the simplest form. For example, we choose to write the top associator, which one can read from Figure ^{\underline{fig:dtstrand}}^{\underline{fig:dtstrand}} 22 is Φ_{13(24)}^{−1}, in terms of c_{13} and c_{1(24)} = c_{12} + c_{14}, since after the puctures α(c_{13}) = a_{31} and α(c_{1(24)}) = a_{41}. This is reflected in Figure ^{\underline{fig:dtstrand2}}^{\underline{fig:dtstrand2}} 23 by drawing only these two arrows as a “place holder” for this associator.

Now observe that the two arguments of α(Φ)_{12(34)} commute: strands 3 and 4 support only tails in this associator, and tails commute (TC). As mentioned before, the value of an associator in the quotient where its two arguments commute is 1, hence α(Φ)_{12(34)} = 1. Next, study α(Φ^{−1})_{134}: the tail of an arrow a_{41} can be “pulled over the top on strand 5” using the VI, CP, and TC relations (in paricular, it commutes with α(ν) and with e^{−a/2}). As a result, a_{41} = a_{31}, meaning in particular that a_{41} commutes with a_{31}, and so α(Φ^{−1})_{134} = 1. Applying the same trick also cancels α(Φ)_{3}24, and α(Φ^{−1})_{234}. In the latter case note that the tail of a_{42} also commutes with the tails of α(R^{−1}). In summary, all associators cancel.

Next we show that α(ν^{1/2})_{(34)}, which remains from n, cancels “with itself”. Since ν is an exponential of wheels, so is α(ν^{1/2}) ∈ A^{w}(↑). (Recall from ^{\underline{Bar-NatanDancso:WKO1}} [WKO1][Section 3.8] that wheels in A^{w} have two possible orientations. For odd wheels these are negatives of each other by the AS relation, for odd wheels they are equal. Hence, α kills odd wheels and multiplies even wheels by 2, as well as orienting them.) Let us write α(ν^{1/2}) as e^{w(x)}, where w(x) is an (even) power series in x with constant term 0: to interpret this as an element of A^{w}(↑), expand it and interpret each monomial x^{k} as a k-wheel on the single strand. Then by the action of unzip, α(ν^{1/2})_{(34)} = e^{w(x_{3}+x_{4})}, where each monomial is interpreted as a cyclic word which is in turn interpreted as a wheel on strands 3 and 4. Now slide this arrow diagram up on strands 3 and 4 to strand 5. Recalling that all associators are dead, and α(R^{−1}) has only tails on strand 3, there is no obstruction to this. At the junction of strands 3, 1 and 5, we need to apply the VI relation. Tails on the punctured strand 1 are zero (TF relation), so each tail slides onto strand 5, whose orientation is compatible with strand 3. In other words we replace x_{3} by x_{5} in the expression. On the other side, tails again slide onto strand 5 but now the orientations are opposite, and hence x_{4} is replaced by −x_{5}. In summary, α(ν^{1/2})_{(34)} = e^{w(x_{5}−x_{5})} = 1.

Finally, move the top exponential e^{−a/2} to strands 3 and 2, using the VI relation at both vertices. The tail of the arrows moves freely from strand 5 to strand 3. The heads commute

DRAFT

⊔^{3}  T_{1}^{1} ^{2} ^{3} ^{4}  T_{2}^{5} ^{6} ^{7}^{,}  T_{2}^{4}  ^{1} ^{2} T_{1}^{1} ^{2} ^{3}  ^{c_{3}}  ^{1} ^{2}  ^{3} T^{1} ^{2} ^{3} ^{4} ^{5} T
Figure 24. Basic planar algebra operations: disjoint union and contraction.  fig:AtomicPAOps

with α(ν), they are killed on strand 4 due to the CP relation, so they slide onto strand 2 but acquire a negative sign. Hence, (e^{−a/2})_{5} = (e^{a/2})_{32}, and this cancels α(R^{−1}).

To summarize, ξ(↑) = α(ν^{1/2}), as claimed. □

Corollary 3.5. The value of the cap is C = α(ν^{1/4}). □

Recall that this lets us define Z^{w}(T) for any T ∈ uT T, in particular the value V of the vertex: namely, Z^{w}(T) is defined to be ξ(T) multiplied by the value C^{−1} at each end of T. Hence, we have defined Z^{w} from Z^{u}, and it is compatible with Z^{u} by design. The next step is to prove that Z^{w} is a homomorphic expansion of w̃TF. The bulk of the argument relies on the following proposition:

Proposition 3.6. Z^{w} is a planar algebra map.

Proof. Planar algebra operations can be written as compositions of two simpler, basic operations: disjoint unions and contractions. For two tangles T_{1} and T_{2}, the disjoint union T_{1} ⊔ T_{2} is the disjoint union of the two tangles where the ends are ordered by declaring that the ordered ends of T_{1} come first, followed by the ordered ends of T_{2}. The contraction operation c_{i} can be applied to any tangle with at least i + 1 ends, and acts by joining the i-th and (i + 1)-st ends of T and re-numbering the rest, resulting in a tangle with two less ends. Both operations are shown in Figure ^{\underline{fig:AtomicPAOps}}^{\underline{fig:AtomicPAOps}} 24.

All we need to show then is that Z^{w} commutes with these two operations, that is, Z^{w}(T_{1} ⊔ T_{2}) = Z^{w}(T_{1}) ⊔ Z^{w}(T_{2}), and Z^{w}(c_{i}(T)) = c_{i}(Z^{w}(T)). Note that the right sides of these equalities make sense: arrow diagrams on uT T skeleta, where Z^{w} takes its values, also form a planar algebra. We first prove the equality for disjoint unions.

We need to compute ξ(T_{1} ⊔ T_{2}), where T_{1} and T_{2} are two uT T -s. The � map applied to a disjoint union of T_{1} and T_{2} is shown in Figure ^{\underline{fig:DisjUni}}^{\underline{fig:DisjUni}} 25. Recall that the trees can be chosen arbitrarily by Lemma ^{\underline{lem:TreeChange}}^{\underline{lem:TreeChange}} 3.1, hence we chose the most convenient trees. Observe that �(T_{1}⊓T_{2}) is the same sKTG as �(T_{1}) and �(T_{2}) inserted into a simpler sKTG, denoted H, as shown in the same figure (up to some orientation switches which don’t impact what follows and hence will be ignored for simplicity). Hence, Z^{u}(�(T_{1} ⊔ T_{2})) is given by inserting Z^{u}(�(T_{1})) and Z^{u}(�(T_{2})) into Z^{u}(H). We could compute Z^{u}(H) using the same algorithm as before, but the good news is we don’t have to: all chords in Z^{u}(H) can be assumed to be located in the rectangle shown in Figure ^{\underline{fig:DisjUni}}^{\underline{fig:DisjUni}} 25. After applying α, both strands on which α(Z^{u}(H)) lives are punctured, and hence it does not survive, i.e., becomes 1 in A^{w} after the punctures. After the subsequent caps and unzips, T_{1} and T_{2} are once again separated, hence we have shown that ξ(T_{1} ⊔ T_{2}) = ξ(T_{1}) ⊔ ξ(T_{2}), and it follows immediately that Z^{w}(T_{1} ⊔ T_{2}) = Z^{w}(T_{1}) ⊔ Z^{w}(T_{2}).

DRAFT

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

Figure 25. The double tree map applied to a disjoint union of uT T -s is the same as inserting the double tree of each individual uT T into the sKTG H. In Z^{u}(H) all chords can be pushed

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

References

AlekseevEnriquezTorossian:ExplicitSolutions [AET] A. Alekseev, B. Enriquez, and C. Torossian, Drinfeld’s associators, braid groups and an explicit solution of the Kashiwara-Vergne equations, Publications Mathématiques de L’IHÉS, 112-1 (2010) 143–189, arXiv:0903.4067.

AlekseevTorossian:KashiwaraVergne [AT] A. Alekseev and C. Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators, Annals

of Mathematics 175 (2012) 415–463, arXiv:0802.4300.

Bar-Natan:GT1 [BN1] D. Bar-Natan, On associators and the Grothendieck-Teichmuller Group I, in Selecta Mathematica,

New Series 4, 183–212, June 1996.

Bar-NatanDancso:KTG [BND1] D. Bar-Natan and Z. Dancso, Homomorphic expansions for knotted trivalent graphs, Journal of Knot

Theory and its Ramifications Vol. 22, No. 1 (2013) arXiv:1103.1896

Bar-NatanGaroufalidisRozanskyThurston:WheelsWheeling [BGRT] D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston, Wheels, wheeling, and the Kontse-

vich integral of the unknot, Israel Journal of Mathematics 119 (2000) 217–237, arXiv:q-alg/9703025.

Bar-NatanLeThurston:TwoApplications [BLT] D. Bar-Natan, T. Q. T. Le, and D. P. Thurston, Two applications of elementary knot theory to Lie al-

gebras and Vassiliev invariants, Geometry and Topology 7-1 (2003) 1–31, arXiv:math.QA/0204311.

KashiwaraVergne:Conjecture [KV] M. Kashiwara and M. Vergne, The Campbell-Hausdorff Formula and Invariant Hyperfunctions, In-

vent. Math. 47 (1978) 249–272.

Bar-NatanDancso:WKO [WKO0] D. Bar-Natan and Z. Dancso, Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne, paper, videos (wClips) and related files at

http://www.math.toronto.edu/~drorbn/papers/WKO/. The arXiv:1309.7155 edition may be

older.

Bar-NatanDancso:WKO1 [WKO1] D. Bar-Natan and Z. Dancso, Finite Type Invariants of W-Knotted Objects I: W-Knots and the

Alexander Polynomial, Alg. Geom. Topology 16 (2016) 1063–1133. arXiv:1405.1956

Bar-NatanDancso:WKO2 [WKO2] D. Bar-Natan and Z. Dancso, Finite Type Invariants of W-Knotted objects II: Tangles, Foams and

the Kashiwara-Vergne Problem to appear in Math. Annalen, arXiv:1405.1955

DoubleTree [WKO3] D. Bar-Natan and Z. Dancso, Finite Type Invariants of w-Knotted Objects III: From Associators to Solutions of the Kashiwara-Vergne problem (self-reference), paper and related files at

http://www.math.toronto.edu/~drorbn/papers/DoubleTree/. The arXiv:????.???? edition may

be older.

Bar-Natan:WKO4 [WKO4] D. Bar-Natan, Finite Type Invariants of w-Knotted Objects IV: Some Computations arXiv:1511.05624

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

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

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

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