\draftcut
\section{w-Tangled Foams} \label{sec:w-foams}
\begin{quote} \small {\bf Section Summary. }
\summaryfoams
\end{quote}
\subsection{The Circuit Algebra of w-Tangled Foams} \label{subsec:wTFo}
In the same manner as we did for tangles, we will present the circuit algebra
of w-tangled foams via its Reidemeister-style diagrammatic description
accompanied by a local topological interpretation.
\begin{definition}\label{def:wTFo}
Let $\glos{\wTFo}$ (where $o$ stands for ``orientable'', to be explained
in Section~\ref{subsec:TheWen}) be the algebraic structure
\[
\wTFo=\CA\!\left.\left.\left\langle
\raisebox{-2mm}{\input{figs/wTFgens.pstex_t}}
\right|
\parbox{1.2in}{\centering w-relations as in Section~\ref{subsubsec:wrels}}
\right|
\parbox{1.2in}{\centering w-operations as in Section~\ref{subsubsec:wops}}
\right\rangle.
\]
Hence $\wTFo$ is the circuit algebra generated by the generators listed
above and described below, modulo the relations described in
Section~\ref{subsubsec:wrels}, and augmented with several ``auxiliary
operations'', which are a part of the algebraic structure of $\wTFo$ but are
not a part of its structure as a circuit algebra, as described in
Section~\ref{subsubsec:wops}. To be more specific, $\wTFo$ is skeleton-graded
where the circuit algebra of skeleta $\calS^o$ is a version of the $\calS$ introduced
in Section~\ref{subsec:CircuitAlgebras}, but with vertices and caps included
(as opposed to only empty circuits).
\parpic[r]{\input{figs/VertexExamples.pstex_t}}
To be completely precise, we have to admit that $\wTFo$ as a circuit
algebra has more generators than shown above. The last two generators
are ``foam vertices'', as will be explained shortly, and exist in all
possible orientations of the three strands. Some examples are shown on
the right. However, in Section~\ref{subsubsec:wops} we will describe the
operation ``orientation switch'' which allows switching the orientation of
any given strand. In the algebraic structure which includes this extra
operation in addition to the circuit algebra structure, the generators
of the definition above are enough.
\end{definition}
\subsubsection{The generators of $\wTFo$}\label{subsubsec:wTFgens}
There is topological meaning to each of the generators of $\wTFo$:
they each stand for a certain local feature of framed knotted ribbon
tubes in $\bbR^4$. As in Section \ref{subsec:TangleTopology},
the tubes are oriented as 2-dimensional surfaces, and also have a
distinguished core with a 1-dimensional orientation (direction).
The crossings are as explained in Section
\ref{subsec:TangleTopology}: the under-strand denotes the ring flying through,
or the ``thin'' tube. Recall that there really are four kinds of crossings,
but in the circuit algebra the two not shown are obtained from the two that are shown
by adding virtual crossings (see Figures~\ref{fig:CrossingTubes} and~\ref{fig:BandCrossings}).
\parpic[r]{\input{figs/Cap.pstex_t}}
A bulleted end denotes a cap on the tube, or a flying ring that shrinks to a point,
as in the figure on the right. For further motivation, in terms of the topological construction of
Satoh's tubing map \cite[Section~\ref{1-subsubsec:TopTube}]{Bar-NatanDancso:WKO1},
the cap means that ``the string is attached to the bottom of the thickened
surface'', as shown in the figure below. We recall that the tubing map is the
composition
$$\gamma\times S^1 \hookrightarrow \Sigma \times [-\epsilon,\epsilon] \hookrightarrow \bbR^4.$$
Here $\gamma$ is a trivalent tangle ``drawn on the virtual surface $\Sigma$'', with caps ending on
$\Sigma \times \{-\epsilon\}$. The first embedding above is the product of this ``drawing'' with an
$S^1$, while the second arises from the unit normal bundle of $\Sigma$ in $\bbR^4$. For each cap $(c, -\epsilon)$ the
tube resulting from Satoh's map has a boundary component $\partial_c=(c,-\epsilon)\times S^1$.
Follow the tubing map by gluing a disc to this boundary component to obtain the capped tube mentioned above.
\begin{center}
\input figs/SatohCap.pstex_t
\end{center}
\parpic[l]{\includegraphics[height=5cm]{figs/TheVertex.ps}}
The last two generators denote singular ``foam vertices''. As the notation
suggests, a vertex can be thought of as ``half of a crossing''. To make
this precise using the flying rings interpretation, the first singular
vertex represents the movie shown on the left: the ring corresponding
to the right strand approaches the ring represented by the left strand
from below, flies inside it, and then the two rings fuse (as opposed to
a crossing where the ring coming from the right would continue to fly
out to above and to the left of the other one). The second vertex is
the movie where a ring splits radially into a smaller and a larger ring,
and the small one flies out to the right and below the big one.
\vspace{5mm}
\parpic[r]{\input{figs/VertexInSurface.pstex_t}}
The vertices can also be interpreted topologically via a natural extension of Satoh's tubing map. For
the first vertex, imagine the broken right strand approaching the continuous left strand directly from below
in a thickened surface, as shown.
The reader might object that there really are four types of vertices (as
there are four types of crossings), and each of these can be viewed as
a ``fuse'' or a ``split'' depending on the strand directions, as shown
in Figure~\ref{fig:VertexTypes}. However, looking at the fuse vertices
for example, observe that the last two of these can be obtained from the
first two by composing with virtual crossings, which always exist in a
circuit algebra.
The sign of a vertex can be defined the same way as the sign of a crossing
(see Section~\ref{subsec:TangleTopology}). We will sometimes refer to
the first generator vertex as ``the positive vertex'' and to the second
one as ``the negative vertex''. We use the band notation for vertices
the same way we do for crossings: the fully coloured band stands for
the thin (inner) ring.
\begin{figure}[h!]
\input{figs/VertexTypes.pstex_t}
\caption{Vertex types in $\wTFo$.}\label{fig:VertexTypes}
\end{figure}
\subsubsection{The relations of $\wTFo$} \label{subsubsec:wrels}
In addition to the usual \Rs, R2, R3, and OC moves of
Figure~\ref{fig:VKnotRels}, more relations are added to describe the
behaviour of the additional features.
\begin{comment}\label{com:MissingTopology}
As before, the relations have local topological explanations, and
we conjecture that they provide a Reidemeister theory for
``w-tangled foams'', that is, knotted ribbon tubes with foam vertices
in $\bbR^4$. In this section we list the relations along with the
topological reasoning behind them. However, for any rigorous purposes below,
$\wTFo$ is studied as a circuit algebra given by the declared generators and relations,
with topology serving only as intuition.
\end{comment}
Recall that topologically, a cap represents a capped tube or equivalently,
flying ring shrinking to a point. Hence, a cap
on the thin (or under) strand can be ``pulled out'' from a crossing,
but the same is not true for a cap on the thick (or over) strand, as
shown below. This is the case for any orientation of the strands. We
denote this relation by \glost{CP}, for Cap Pull-out.
\[ \input{figs/CapRel.pstex_t} \]
The Reidemeister 4 relations assert that a strand can be moved under
or over a crossing, as shown in the picture below. The ambiguously
drawn vertices in the picture denote a vertex of any kind (as described
in Section~\ref{subsubsec:wTFgens}), and the strands can be oriented
arbitrarily. The local topological (tube or flying ring) interpretations
can be read from the pictures below. These relations will be denoted
\glost{R4}.
\begin{center}
\input{figs/R4.pstex_t}
\end{center}
\subsubsection{The auxiliary operations of $\wTFo$} \label{subsubsec:wops}
The circuit algebra $\wTFo$ is equipped with several extra operations.
The first of these is the familiar orientation switch. We will, as mentioned in
Section~\ref{subsec:TangleTopology}, distinguish between switching both the 2D and 1D orientations, or just the
strand (1D) direction.
Topologically {\it orientation switch}, denoted $\glos{S_e}$, is the
switch of both orientations of the strand $e$. Diagrammatically (and this
is the definition) $S_e$ is the operation which reverses the orientation
of a strand in a $\wTFo$ diagram. The reader can check that when applying
Satoh's tubing map, this amounts to reversing both the direction and
the 2D orientation of the tube arising from the strand.
\parpic[r]{\input{figs/Adjoint.pstex_t}}
The operation which, in topology world, reverses a tube's direction
but not its 2D orientation is called {\it ``adjoint''}, and denoted by
$\glos{A_e}$. This is slightly more intricate to define rigorously in terms
of diagrams. In addition to reversing the direction of the strand $e$
of the $\wTFo$ diagram, $A_e$ also locally changes each crossing of $e$
{\em over} (but not under!) another strand by adding two virtual crossings, as shown
on the right. We recommend for the reader to convince themselves that
this indeed represents a direction switch in topology after reading
Section~\ref{subsec:TheWen}.
\begin{remark}\label{rem:SwitchingVertices}
As an example, let us observe how the negative generator vertex
can be obtained from the positive generator vertex by adjoint
operations and composition with virtual crossings, as shown in
Figure~\ref{fig:VertexSwitch}. Note that also all other vertices can
be obtained from the positive vertex via orientation switch and adjoint
operations and composition by virtual crossings.
As a small exercise, it is worthwhile to convince ourselves of the effect
of orientation switch operations on the {\em band picture}. For example,
replace $A_1A_2A_3$ by $S_1S_2S_3$ in figure \ref{fig:VertexSwitch}.
In the strand diagram, this will only reverse the direction of the
strands. The reader can check that in the band picture not only the
arrows will reverse but also the blue band will switch to be on top of
the red band.
\end{remark}
\begin{figure}
\input figs/VertexSwitch.pstex_t
\caption{Switching strand orientations at a vertex. The adjoint operation
only switches the tube direction, hence in the \emph{band picture}
only the arrows change. To express this vertex in terms of the negative
generating vertex in strand notation, we use a virtual crossing (see
Figure~\ref{fig:VertexTypes}).}\label{fig:VertexSwitch}
\end{figure}
\begin{comment}\label{com:wTFFraming}
Framings were discussed in Section \ref{subsec:TangleTopology}, but have not played
a significant role so far, except to explain the lack of a Reidemeister 1
relation. We now need to discuss framings in order to provide a topological
explanation for the unzip (tube doubling) operation.
In the local topological interpretation of $\wTFo$, strands represent ribbon-knotted tubes with foam vertices,
which are also equipped with a framing, arising from the blackboard framing of the strand diagrams via Satoh's tubing map.
Strand doubling is the operation of doubling a tube by ``pushing it off itself slightly'' in the framing direction,
as shown in Figure \ref{fig:BandUnzip}.
Recall that ribbon knotted tubes have a ``filling'', with only ``ribbon''
self-intersections
\cite[Section~\ref{1-subsubsec:ribbon}]{Bar-NatanDancso:WKO1}. When we
double a tube, we want this ribbon property to be preserved. This is
equivalent to saying that the ring obtained by pushing off any given girth
of the tube in the framing direction is not linked with the original tube,
which is indeed the case.
\parpic[r]{$\input{figs/DetailedVertex.pstex_t}$}
Framings arising from the blackboard framing of strand diagrams via Satoh's tubing map
always match at the vertices, with the normal vectors pointing either directly towards or
away from the center of the singular ring. Note that the orientations of the three tubes may or may not match.
An example of a vertex with the orientations and framings shown is on the right. Note that the
framings on the two sides of each band are mirror images of each other, as they should be.
\end{comment}
\begin{figure}
\input{figs/BandUnzip.pstex_t}
\caption{Unzipping a tube, in band notation with orientations and framing marked.}\label{fig:BandUnzip}
\end{figure}
Unzip, or tube doubling is perhaps the most interesting of the auxiliary $\wTFo$ operations. As mentioned above, topologically
this means pushing the tube off
itself slightly in the framing direction. At each of the vertices at the two ends of the doubled tube there are two tubes
to be attached to the doubled tube. At each end, the normal vectors pointed either directly towards or away from the center,
so there is an ``inside'' and an ``outside'' ending ring. The two tubes to be attached also come as an ``inside'' and an
``outside'' one, which defines which one to attach to which. An example is shown in Figure \ref{fig:BandUnzip}. Unzip
can only be done if the 1D and 2D orientations match at both ends.
\parpic[r]{\input{figs/StringUnzip.pstex_t}}
To define unzip rigorously, we must talk only of strand diagrams. The
natural definition is to let $\glos{u_e}$ double the strand $e$ using the
blackboard framing, and then attach 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. This is a somewhat
artificial condition which we impose to get equations equivalent to
the~\cite{AlekseevTorossian:KashiwaraVergne} equations.
A related operation, {\it disk unzip}, is unzip done on a capped strand, pushing the tube off in the direction of the framing
(in diagrammatic world, in the direction of the blackboard framing), as
before. An example in the line and band notations (with the framing suppressed) is shown below.
\begin{center}
\input{figs/CapUnzip.pstex_t}
\end{center}
Finally, we allow deletion $d_e$ of ``long linear'' strands, meaning strands that do not end in a vertex on either side.
To summarize,
\[
\wTFo=\CA\!\left.\left.\left\langle
\raisebox{-2mm}{\input{figs/wTFgens.pstex_t}}
\right|
\parbox{1.2in}{\centering \Rs, R2, R3, R4, OC, CP}
\right|
\parbox{1.2in}{\centering $S_e, A_e, u_e, d_e$}
\right\rangle.
\]
The goal, as before, is to construct a homomorphic expansion for $\wTFo$. However, first we need to understand its target space,
the associated graded structure $\grad \wTFo$.
\draftcut
\subsection{The Associated Graded Structure} \label{subsec:fgrad}
Mirroring the previous section, we describe the associated graded
$\glos{\calA^{sw}}$ of $\wTFo$ and its ``full version'' $\glos{\calA^w}$
as circuit algebras on certain generators modulo a number of
relations. From now on we will write $\glos{A^{(s)w}}$ to mean ``$\calA^{w}$
and/or $\calA^{sw}$''.
\[ \calA^{(s)w}=\CA\!\left.\left.\left\langle
\raisebox{-2mm}{\input{figs/wTFprojgens.pstex_t}}
\right|
\parbox{1.2in}{\centering relations as in Section~\ref{subsubsec:wTFGradRels}}
\right|
\parbox{1.2in}{\centering operations as in Section~\ref{subsubsec:wTFGradOps}}
\right\rangle.
\]
In other words, $\calA^{(s)w}$ are the circuit algebras of arrow diagrams on trivalent (or foam) skeletons with
caps; that is, skeleta are elements of $\calS^o$ as in Definition \ref{def:wTFo}.
Note that all but the first of the generators are skeleton features (of degree 0), and that the single arrow is
the only generator of degree $1$. As for the generating vertices, the same remark applies as in Definition \ref{def:wTFo},
that is, there are more vertices with all possible strand orientations needed to generate $\calA^{(s)w}$ as circuit algebras.
\subsubsection{The relations of $\calA^{(s)w}$}\label{subsubsec:wTFGradRels}
In addition to the usual $\aft$ and TC
relations (see Figure \ref{fig:TCand4T}), as well as RI in the case of $\calA^{sw}=\calA^w/RI$,
diagrams in $\calA^{(s)w}$ satisfy the following additional relations:
{\it Vertex invariance}, denoted by \glost{VI}, are relations arising
the same way as $\aft$ does, but with the participation of a vertex as
opposed to a crossing:
\begin{center}
\input figs/VI.pstex_t
\end{center}
The other end of the arrow is in the same place throughout the relation, somewhere outside the picture
shown. The signs are positive whenever the strand on which the arrow ends is directed towards the vertex,
and negative when directed away. The ambiguously drawn vertex means any kind of vertex, but the same one throughout.
\parpic[r]{\input{figs/CapHeads.pstex_t}}
The CP relation (a cap can be pulled out from under a strand but not from
over, Section \ref{subsubsec:wrels}) implies that arrow heads near a cap
are zero, as shown on the right. Denote this relation also by
\glost{CP}. (Also note that a tail near a cap is not set to zero.)
As in the previous sections, and in particular in Definition~\ref{def:wJac}, we define a
``w-Jacobi diagram'' (or just ``arrow diagram'') on a foam
skeleton by allowing trivalent chord vertices. Denote the circuit algebra of formal
linear combinations of arrow diagrams by $\calA^{(s)wt}$. We have the following bracket-rise theorem:
\begin{theorem} The obvious inclusion of diagrams induces a circuit
algebra isomorphism $\calA^{(s)w}\cong\calA^{(s)wt}$. Furthermore, the $\aAS$
and $\aIHX$ relations of Figure~\ref{fig:aIHX} hold in $\calA^{(s)wt}$.
\end{theorem}
\begin{proof} Same as the proof of Theorem~\ref{thm:BracketRise}. \qed
\end{proof}
As in Section~\ref{subsec:vw-tangles}, the primitive elements of
$\calA^{(s)w}$ are connected diagrams (that is, connected even with the skeleton removed), namely trees and wheels. Before
moving on to the auxiliary operations of $\calA^{(s)w}$, let us make
two useful observations:
\begin{lemma}\label{lem:CapIsWheels}
$\calA^w(\raisebox{-1mm}{\input{figs/SmallCap.pstex_t}})$, the part of
$\calA^w$ with skeleton $\raisebox{-1mm}{\input{figs/SmallCap.pstex_t}}$,
is isomorphic as a vector space to the completed polynomial
algebra freely generated by wheels $w_k$ with $k \geq 1$. Likewise
$\calA^{sw}(\raisebox{-1mm}{\input{figs/SmallCap.pstex_t}})$, except here
$k \geq 2$.
\end{lemma}
\begin{proof}
Any arrow diagram with an arrow head at its top is zero by the Cap Pull-out (CP) relation. If $D$ is an arrow
diagram that has a head somewhere on the skeleton but not at the top, then one can use repeated $\aSTU$ relations
to commute the head to the top at the cost of diagrams with one fewer skeleton head.
Iterating
this procedure, we can get rid of all arrow heads, and hence write $D$ as a linear combination of
diagrams having no heads on the skeleton. All connected components of such diagrams are wheels.
To prove that there are no relations between wheels in $\calA^{(s)w}(\raisebox{-1mm}{\input{figs/SmallCap.pstex_t}})$,
let $S_L\colon \calA^{(s)w}(\uparrow_1) \to \calA^{(s)w}(\uparrow_1)$
(resp. $S_R$) be the map that sends an arrow diagram to the sum of all ways of dropping one left (resp. right) arrow
(on a vertical strand, left means down and right means up). Define
$$\glos{F}:=\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}D_R^k(S_L+S_R)^k,$$
where $D_R$ is a short right arrow.
We leave it as an exercise for the reader to check that $F$ is a bi-algebra homomorphism that kills diagrams with an arrow head at the top
(i.e., CP is in the kernel of $F$), and $F$ is injective on wheels. This concludes the proof.
\qed
\end{proof}
\begin{lemma}
$\calA^{(s)w}(Y)=\calA^{(s)w}(\uparrow_2)$, where $\calA^{(s)w}(Y)$
stands for the space of arrow diagrams whose skeleton is a $Y$-graph (i.e., a vertex) with
any orientation of the strands, and as before $\calA^{(s)w}(\uparrow_2)$
is the space of arrow diagrams on two strands.
\end{lemma}
\begin{proof}
We can use the vertex invariance (VI) relation to push all arrow heads and tails from the ``trunk'' of the vertex to the other two strands.
\qed
\end{proof}
\subsubsection{The auxiliary operations of $\calA^{(s)w}$}\label{subsubsec:wTFGradOps}
Recall from Section \ref{subsec:TangleTopology} that the orientation switch $S_e$ (i.e. changing both the $1D$ and $2D$ orientations of a strand) always changes
the sign of a crossing involving the strand $e$. Hence, letting $S$ denote any foam (trivalent) skeleton, the induced arrow diagrammatic
operation is a map $S_e\colon \calA^{(s)w}(S) \to \calA^{(s)w}(S_e(S))$ which acts by multiplying
each arrow diagram by $(-1)$ raised to the number of arrow endings on $e$ (counting both heads and tails).
The adjoint operation $A_e$ (i.e. switching only the strand direction), on the other hand, only changes the sign of a crossing when the
strand being switched is the under- (or through) strand. (See section \ref{subsec:TangleTopology} for pictures and explanation.)
Therefore, the arrow diagrammatic $A_e$ acts by switching the direction
of $e$ and multiplying each arrow diagram by $(-1)$ raised to the number
of {\it arrow heads} on $e$. Note that in $\calA^{(s)w}(\uparrow_n)$
taking the adjoint on every strand gives the adjoint map of Definition
\ref{def:Adjoint}.
\parpic[r]{\input{figs/Unzip.pstex_t}}
The arrow diagram operations induced by unzip and disc unzip (both to be denoted $u_e$, and interpreted appropriately according to whether the
strand $e$ is capped) are maps $u_e\colon \calA^{(s)w}(S) \to \calA^{(s)w}(u_e(S))$, where each arrow ending (head or tail) on $e$ is mapped to
a sum of two arrows, one ending on each of the new strands, as shown on the right. In other words, if in an arrow diagram $D$ there are $k$ arrow
ends on $e$, then $u_e(D)$ is a sum of $2^k$ arrow diagrams.
The operation induced by deleting the long linear strand $e$ is the map $d_e\colon \calA^{(s)w}(S) \to \calA^{(s)w}(d_e(S))$ which kills arrow diagrams with
any arrow ending (head or tail) on $e$, and leaves all else unchanged, except with $e$ removed.
\begin{definition}In summary,
\[ \calA^{(s)w}=\CA\!\left.\left.\left\langle
\raisebox{-2mm}{\input{figs/wTFprojgens.pstex_t}}
\right|
\parbox{1.2in}{\centering $\aft$, TC, VI, CP, (RI for $\calA^{sw}$)}
\right|
\parbox{1.2in}{\centering $S_e, A_e, u_e, d_e$}
\right\rangle.
\]
\end{definition}
\draftcut
\subsection{The homomorphic expansion}\label{subsec:wTFExpansion}
The following is one of the main theorems of this paper:
\begin{theorem}[Proof in Section~\ref{subsec:EqWithAT}]
\label{thm:ATEquivalence} There exists a group-like\footnote{The
formal definition of the group-like property is along the lines of
\cite[Section~\ref{1-par:Delta}]{Bar-NatanDancso:WKO1}. In practice, it means
that the $Z$-values of the vertices, crossings, and cap (denoted $V$,
$R$ and $C$ below) are exponentials of linear combinations of
connected diagrams.}
homomorphic expansion for $\wTFo$, i.e. a group-like expansion $Z\colon
\wTFo \to \calA^{sw}$ which is a map of circuit algebras and also
intertwines the auxiliary operations of $\wTFo$ with their arrow
diagrammatic counterparts. In fact,
there is a bijection
between the set of solutions $(F,a)$ of the generalized KV problem (see
Section~\ref{subsec:EqWithAT}) and the set of homomorphic expansions
for $\wTFo$ which do not contain local arrows\footnote{For a detailed
explanation of this minor point see the third paragraph of the proof.}
in the value $V$ of the vertex.
\end{theorem}
Since both $\wTFo$ and $\calA^{sw}$ are circuit algebras defined
by generators and relations, when looking for a suitable $Z$
all we have to do is to find values for each of the generators
of $\wTFo$ so that these satisfy (in $\calA^{sw}$) the equations
which arise from the relations in $\wTFo$ and the homomorphicity
requirement. In this section we derive these equations and in the next
section we show
that they are equivalent to the Alekseev-Torossian version of the
Kashiwara-Vergne equations \cite{AlekseevTorossian:KashiwaraVergne}. In
\cite{AlekseevEnriquezTorossian:ExplicitSolutions} Alekseev Enriquez
and Torossian construct explicit solutions to these equations using
associators. In~\cite{Bar-NatanDancso:WKO3} we will interpret and independentlt prove this
result in our context of homomorphic expansions for w-tangled foams.
Let $\glos{R}:=Z(\overcrossing) \in \calA^{sw}(\uparrow_2)$. It
follows from the Reidemeister 2 relation that
$Z(\undercrossing)=(R^{-1})^{21}$. As discussed in Sections
\ref{subsec:vw-tangles} and \ref{subsec:UniquenessForTangles},
Reidemeister 3 with group-likeness and homomorphicity
implies that $R=e^a$, where $a$ is a single arrow
pointing from the over to the under strand. Let
$\glos{C}:=Z(\raisebox{-1mm}{\input{figs/SmallCap.pstex_t}})\in
\calA^{sw}(\raisebox{-1mm}{\input{figs/SmallCap.pstex_t}})$.
By Lemma \ref{lem:CapIsWheels}, we know that
$C$ is made up of wheels only. Finally, let
$\glos{V}=\glos{V^+}:=Z(\raisebox{-1mm}{\input{figs/PlusVertex.pstex_t}})\in
\calA^{sw}(\raisebox{-1mm}{\input{figs/PlusVertex.pstex_t}})\cong
\calA^{sw}(\uparrow_2)$, and
$\glos{V^-}:=Z(\raisebox{-1mm}{\input{figs/MinusVertex.pstex_t}})\in
\calA^{sw}(\raisebox{-1mm}{\input{figs/MinusVertex.pstex_t}})\cong
\calA^{sw}(\uparrow_2)$.
Before we translate each of the relations of Section \ref{subsubsec:wrels} to equations let us slightly extend the notation used in
Section \ref{subsec:UniquenessForTangles}. Recall that $R^{23}$, for instance, meant ``$R$ placed on strands 2 and 3''. In this section
we also need notation such as $R^{(23)1}$, which means ``$R$ with its first strand doubled, placed on strands 2, 3 and 1''.
Now on to the relations, note that Reidemeister 2 and 3 and Overcrossings Commute have already been dealt with. Of the two Reidemeister 4
relations, the first one induces an equation that is automatically satisfied. Pictorially, the equation
looks as follows:
\begin{center}
\input{figs/R4ToEquation.pstex_t}
\end{center}
In other words, we obtained the equation
$$V^{12}R^{3(12)}=R^{32}R^{31}V^{12}.$$
However, observe that by the ``head-invariance'' property of arrow diagrams (Remark \ref{rem:HeadInvariance})
$V^{12}$ and $R^{3(12)}$ commute on the left hand side. Hence the left hand side equals $R^{3(12)}V^{12}=R^{32}R^{31}V^{12}$.
Also, $R^{3(12)}=e^{a^{31}+a^{32}}=e^{a^{32}}e^{a^{31}}=R^{32}R^{31}$, where the second step is due to the fact that
$a^{31}$ and $a^{32}$ commute. Therefore, the equation is true independently of the choice of $V$.
We have no such luck with the second Reidemeister 4 relation, which, in the same manner as in the paragraph above,
translates to the equation
\begin{equation}\label{eq:HardR4}
V^{12}R^{(12)3}=R^{23}R^{13}V^{12}.
\end{equation}
There is no ``tail invariance'' of arrow diagrams, so $V$ and $R$ do not commute on the left hand side; also, $R^{(12)3}\neq R^{23}R^{13}$.
As a result, this equation puts a genuine restriction on the choice of $V$.
The Cap Pull-out (CP) relation translates to the equation $R^{12}C^2=C^2$. This is true independently of the choice of $C$: by head-invariance,
$R^{12}C^2=C^2R^{12}$. Now $R^{12}$ is just below the cap on strand $2$, and the cap ``kills heads'', in other words, every term of $R^{12}$
with an arrow head at the top of strand $2$ is zero. Hence, the only surviving term of $R^{12}$ is $1$ (the empty diagram), which makes the
equation true.
The homomorphicity of the orientation switch operation was used to prove the uniqueness of $R$ in
Theorem \ref{thm:Tangleuniqueness}. The homomorphicity of the adjoint leads to the equation
$V_-=A_1A_2(V)$ (see Figure \ref{fig:VertexSwitch}), eliminating $V_-$ as an unknown.
Note that we also silently assumed these homomorphicity properties when we did not introduce
32 different values of the vertex depending on the strand orientations.
Homomorphicity of the (annular) unzip operation leads to an equation for $V$, which we are going to refer to as ``unitarity''. This
is illustrated in the figure below. Recall that $A_1$ and $A_2$ denote the adjoint (direction switch) operation on strand 1
and 2, respectively.
\begin{center}
\input figs/Unitarity.pstex_t
\end{center}
Reading off the equation, we have
\begin{equation}\label{eq:unitarity}
V\cdot A_1A_2(V)=1.
\end{equation}
\parpic[r]{\input{figs/CapEqn.pstex_t}}\picskip{5}
Homomorphicity of the disk unzip leads to an equation for $C$ which
we will refer to as the ``cap equation''. The translation from
homomorphicity to equation is shown in the figure on the right. $C$,
as we introduced before, denotes the $Z$-value of the cap. Hence, the
cap equation reads
\begin{equation}\label{eq:CapEqn}
V^{12}C^{(12)}=C^1C^2 \qquad\text{in}\quad
\calA^{sw}(\raisebox{-1mm}{\input{figs/SmallCap.pstex_t}}_2)
\end{equation}
The homomorphicity of deleting long strands does not lead to an equation on its own, however it was
used to prove the uniqueness of $R$ (Theorem \ref{thm:Tangleuniqueness}).
To summarize, we have reduced the problem of finding a homomorphic
expansion $Z$ to finding the $Z$-values of the (positive) vertex and
the cap, denoted $V$ and $C$, subject to three equations: the ``hard
Reidemeister 4'' equation (\ref{eq:HardR4}); ``unitarity of V'' equation
(\ref{eq:unitarity}); and the ``cap equation'' (\ref{eq:CapEqn}).
\draftcut
\subsection{The equivalence with the Alekseev-Torossian equations}
\label{subsec:EqWithAT}
First let us recall Alekseev and Torossian's
formulation of the generalized Kashiwara-Vergne problem
(see~\cite[Section~5.3]{AlekseevTorossian:KashiwaraVergne}):
{\bf Generalized KV problem:} Find an element $\glos{F}\in \TAut_2$ with the properties
\begin{equation}\label{eq:ATKVEqns}
F(x+y)=\log(e^xe^y), \text{ and } j(F)\in \im(\tilde{\delta}).
\end{equation}
Here $\tilde{\delta}\colon \attr_1 \to \attr_2$ is defined by $(\tilde{\delta}a)(x,y)=a(x)+a(y)-a(\log(e^{x}e^{y}))$,
where $\attr_2$ is generated by cyclic words in the letters $x$ and $y$. (See
\cite{AlekseevTorossian:KashiwaraVergne}, Equation (8)). Note that an element of $\attr_1$ is a power series in one variable
with no constant term.
In other words, the second condition says that there exists
$a \in \attr_1$ such that $jF=a(x)+a(y)-a(\log(e^{x}e^{y}))$.
\noindent{\em Proof of Theorem~\ref{thm:ATEquivalence}.}
We have reduced the problem of finding a homomorphic expansion to finding group-like solutions $V$ and $C$ to the hard Reidemeister 4 equation (\ref{eq:HardR4}),
the unitarity equation (\ref{eq:unitarity}), and the cap equation (\ref{eq:CapEqn}).
Suppose we have found such solutions and write $V=e^be^{uD}$, where $b \in \tr_2^s$, $D \in \tder_2\oplus \fraka_2$,
and where $u$ is the map $u\colon \tder_2 \to \calA^{sw}(\uparrow_2)$ which plants the head of a tree
above all of its tails, as introduced in Section \ref{subsec:ATSpaces}. $V$ can be written in this form without loss of generality because wheels can always be commuted to the
bottom of a diagram (at the possible cost of more wheels). Furthermore, $V$ is group-like and hence it can be written in exponential form. Similarly, write
$C=e^c$ with $c \in \attr_1^s$.
Note that $u(\fraka_2)$ is central in $\calA^{sw}(\uparrow_2)$ and
that replacing a solution $(V,C)$ by $(e^{u(a)}V, C)$ for any $a \in
\fraka_2$ does not interfere with any of the equations (\ref{eq:HardR4}),
(\ref{eq:unitarity}) or (\ref{eq:CapEqn}). Hence we may assume that $D$
does not contain any single arrows, that is, $D \in \tder_2$. Also, a
solution $(V,C)$ in $\calA^{sw}$ can be lifted to a solution in $\calA^w$
by simply setting the degree one terms of $b$ and $c$ to be zero. It is
easy to check that this $b \in \attr_2$ and $c \in \attr_1$ along with $D$
still satisfy the equations. (In fact, in $\calA^w$ (\ref{eq:unitarity})
and (\ref{eq:CapEqn}) respectively imply that $b$ is zero in degree 1,
and that the degree 1 term of $c$ is arbitrary, so we may as well assume
it to be zero.) In light of this we declare that $b\in \attr_2$ and $c
\in \attr_1$.
The hard Reidemeister 4 equation (\ref{eq:HardR4}) reads $V^{12}R^{(12)3}=R^{23}R^{13}V^{12}$. Denote the arrow from strand 1 to strand 3 by $x$, and the
arrow from strand 2 to strand 3 by $y$. Substituting the known value for $R$ and rearranging, we get
$$e^be^{uD}e^{x+y}e^{-uD}e^{-b}=e^ye^x.$$ Equivalently, $e^{uD}e^{x+y}e^{-uD}=e^{-b}e^ye^xe^b.$ Now on the right side there are only tails on the
first two strands, hence $e^b$ commutes with $e^ye^x$, so $e^{-b}e^b$ cancels. Taking logarithm of both sides we obtain
$e^{uD}(x+y)e^{-uD}=\log e^ye^x$. Now for notational alignment with \cite{AlekseevTorossian:KashiwaraVergne} we switch strands 1 and 2, which exchanges
$x$ and $y$ so we obtain:
\begin{equation}\label{eq:HardR4Translated}
e^{uD^{21}}(x+y)e^{-uD^{21}}=\log e^xe^y.
\end{equation}
The unitarity of $V$ (Equation (\ref{eq:unitarity})) translates to $1=e^be^{uD}(e^be^{uD})^*,$ where $*$ denotes the adjoint map (Definition \ref{def:Adjoint}). Note that the adjoint switches
the order of a product and acts trivially on wheels. Also, $e^{uD}(e^{uD})^*=J(e^D)=e^{j(e^D)}$, by Proposition \ref{prop:Jandj}.
So we have $1=e^be^{j(e^D)}e^b$. Multiplying by $e^{-b}$ on the right and by $e^b$ on the left, we get $1=e^{2b}e^{j(e^D)}$, and again by switching strand 1 and 2 we arrive at
\begin{equation}\label{eq:UnitarityTranslated}
1=e^{2b^{21}}e^{j(e^{D^{21}})}.
\end{equation}
As for the cap equation, if $C^1=e^{c(x)}$ and $C^2=e^{c(y)}$, then $C^{12}=e^{c(x+y)}$. Note that wheels
on different strands commute, hence $e^{c(x)}e^{c(y)}=e^{c(x)+c(y)}$, so the cap equation reads $$e^be^{uD}e^{c(x+y)}=e^{c(x)+c(y)}.$$ As this equation lives
in the space of arrow diagrams on two \emph{capped} strands, we can multiply the left side on the right by $e^{-uD}$: $uD$ has its head at the top, so it
is 0 by the Cap relation, hence $e^{uD}=1$ near the cap. Hence, $$e^be^{uD}e^{c(x+y)}e^{-uD}=e^{c(x)+c(y)}.$$
\parpic[r]{\input{figs/Sigma.pstex_t}}
On the right side of the equation above \linebreak $e^{uD}e^{c(x+y)}e^{-uD}$ reminds us of Equation (\ref{eq:HardR4Translated}), however we cannot use (\ref{eq:HardR4Translated})
directly as we are working in a different space now. In particular, $x$ there meant an arrow from strand 1 to strand 3, while here it means a one-wheel on (capped)
strand 1, and similarly for $y$. Fortunately, there is a map $\sigma\colon \calA^{sw}(\uparrow_3) \to \calA^{sw}(\raisebox{-1mm}{\input{figs/SmallCap.pstex_t}}_2)$,
where $\sigma$ ``closes the third strand and turns it into a chord (or internal) strand, and caps the first two strands'', as shown on the right. This map is
well defined (in fact, it kills almost all relations, and turns one $\aSTU$ into an $\aIHX$). Under this map, using our abusive notation, $\sigma(x)=x$ and
$\sigma(y)=y$.
Now we can apply Equation (\ref{eq:HardR4Translated}) to get $e^{uD}e^{c(x+y)}e^{-uD}=e^{c(\log e^y e^x)}$.
Substituting this into the cap equation we obtain
$e^be^{c(\log e^y e^x)}=e^{c(x)+c(y)}$, which, using that tails commute, implies
$b=c(x)+c(y)-c(\log e^y e^x)$. Switching strands 1 and 2, we obtain
\begin{equation}\label{eq:CapEqnTranslated}
b^{21}=c(x)+c(y)-c(\log e^x e^y)
\end{equation}
In summary, we can use $(V,C)$ to produce $F:=e^{D^{21}}$ (sorry\footnote{%
We apologize for the annoying $2\leftrightarrow 1$ transposition in this equation,
which makes some later equations, especially~\eqref{eq:ATPhiandV},
uglier than they could have been. There is no depth here, just
mis-matching conventions between us and Alekseev-Torossian.
})
and $a:=-2c$ which satisfy the Alekseev-Torossian equations
(\ref{eq:ATKVEqns}), as follows: $e^{D^{21}}$ acts on $\lie_2$ by conjugation
by $e^{uD^{21}}$, so the first part of (\ref{eq:ATKVEqns})
is implied by (\ref{eq:HardR4Translated}). The second half of
(\ref{eq:ATKVEqns}) is true due to (\ref{eq:UnitarityTranslated}) and
(\ref{eq:CapEqnTranslated}).
On the other hand, suppose that we have found $F\in \TAut_2$ and $a \in \tr_1$ satisfying (\ref{eq:ATKVEqns}).
Then set $D^{21}:=\log F$, $b^{21}:=\frac{-j(e^{D^{21}})}{2}$,
and $c \in \tilde{\delta}^{-1}(b^{21})$, in particular $c=-\frac{a}{2}$ works. Then $V=e^be^{uD}$ and $C=e^c$ satisfy the equations for
homomorphic expansions (\ref{eq:HardR4}), (\ref{eq:unitarity})
and (\ref{eq:CapEqn}).
Furthermore, the two maps between solutions of the KV problem and homomorphic expansions for $\wTFo$ defined in the last two paragraphs are
obviously inverses of each other, and hence they provide a bijection between these sets as stated.\qed
\draftcut
\subsection{The wen}\label{subsec:TheWen}
A topological feature of w-tangled foams which we excluded from the theory
so far is the wen $\glos{w}$. The wen is a Klein bottle cut apart (as mentioned
in \cite[Section~\ref{1-subsubsec:NonHorRings}]{Bar-NatanDancso:WKO1}); in other words it amounts
to changing the 2D orientation of a tube, as shown in the picture below:
\begin{center}
\input{figs/Wen2.pstex_t}
\end{center}
In this section we study the circuit algebra of w-Tangled Foams with the wen included as a generator,
and denote this space by $\glos{\wTF}$. The wen is also added to the circuit algebra of skeleta.
We will find that homomorphic expansions for $\glos{\wTF}$ are
in bijection with solutions to the KV problem with ``even Duflo function'', as explained below.
\subsubsection{The relations and auxiliary operations of $\wTF$.}
\label{subsubsec:WenRels}
Adding the wen as a generator means we have to impose additional relations
involving the wen to keep our topological heuristics intact, as follows:
The interaction of a wen and a crossing is described by the following
relation
(cf.~\cite[Section~\ref{1-subsubsec:NonHorRings}]{Bar-NatanDancso:WKO1}):
\begin{center}
\input figs/FlipRels2.pstex_t
\end{center}
To explain this relation note that in flying ring language, a wen is a ring that flips over. It does not matter whether ring B flips first and
then flies through ring A or vice versa. However, the movies in which ring A first flips and then ring B flies through it,
or B flies through A first and then A flips differ in the fly-through direction of B through A, which is cancelled by virtual crossings,
as in the figure above. We will refer to these relations as the Flip Relations, and abbreviate them by \glost{FR}.
A double flip is homotopic to no flip, in other words two consecutive wens
equal no wen. Let us denote this relation by $\glos{W^2}$, for Wen
squared. Note that this relation explains why there are no ``left and
right wens''.
\parpic[r]{$\input{figs/CapWen.pstex_t}$}
A cap can slide through a wen, hence a capped wen disappears, as shown
on the right, to be denoted \glost{CW}.
\vspace{3mm}
\parpic[l]{\parbox{1.6in}{
\input{figs/LongUparrow.pstex_t}
\hspace{3mm}
\includegraphics[angle=90, height=5 cm]
{figs/TheTwistedVertex.ps}
\hspace{3mm}\raisebox{2.3cm}{$\leftrightarrow$}
\raisebox{-2mm}{\includegraphics[angle=90, height=5 cm]
{figs/TheNegativeVertex.ps}}
\newline
\input{figs/VertexWen.pstex_t}
}}
The last wen relation describes the interaction of wens and
vertices. Recall that there are four types of vertices with the same
strand orientation: among the bottom two bands (in the pictures on
the left) there is a non-filled and a filled band (corresponding to
over/under in the strand diagrams), meaning the ``large'' ring and the
``small'' one which flies into it before they merge. Furthermore, there
is a top and a bottom band (among these bottom two, with apologies for the
ambiguity in overusing the word bottom): this denotes the fly-in direction
(flying in from below or from above). Conjugating a vertex by three wens
switches the top and bottom bands, as shown in the figure on the left:
if both rings flip, then merge, and then the merged ring flips again,
this is homotopic to no flips, except the fly-in direction (from below
or from above) has changed. We are going to denote this relation by
\glost{TV}, for ``twisted vertex''.
The auxiliary operations are the same as for $\wTFo$: orientation
switches, adjoints, deletion of long linear strands, cap unzips
and unzips\footnote{%
We need not specify how to unzip an edge $e$ that carries a wen. To unzip
such $e$, first use the TV relation to slide the wen off $e$.}.
\begin{definition}Summarizing the above, we have
\[
\wTF=\CA\!\left.\left.\left\langle
\raisebox{-2mm}{\input{figs/wTFgensWen.pstex_t}}
\right|
\parbox{2in}{\centering \Rs, R2, R3, R4, OC, CP, FR, $W^2$, CW, TV}
\right|
\parbox{1in}{\centering $S_e, A_e, u_e, d_e$}
\right\rangle.
\]
\end{definition}
\subsubsection{The Associated Graded Structure}\label{subsubsec:AwWen}
The associated graded structure of $\wTF$ (still denoted $\calA^{sw}$) is the same as the associated graded
for $\wTFo$ but with the wen added as a generator (a degree 0 skeleton feature), and with extra relations
describing the behaviour of the wen. Of course, the relations describing the interaction of wens
with the other skeleton features ($W^2$, TV, and CW) still apply, as well as the old $\aft$, TC, VI, CP, and RI relations.
In addition, the Flip Relations FR imply that wens ``commute'' with
arrow heads, but ``anti-commute'' with tails. We also call these \glost{FR}
relations:
\begin{center}
\input figs/WenRel.pstex_t
\end{center}
That is,
\[ \calA^{sw}=\CA\!\left.\left.\left\langle
\raisebox{-2mm}{\input{figs/wTFprojgensWen.pstex_t}}
\right|
\parbox{1.8in}{\centering $W^2$, TW, CW, $\aft$, TC, VI, CP, RI, FR}
\right|
\parbox{1in}{\centering $S_e, A_e, u_e, d_e$}
\right\rangle.
\]
\subsubsection{The homomorphic expansion}\label{subsubsec:ZwithWen}
The goal of this section is to prove that there exists a homomorphic expansion $Z$ for $\wTF$. This involves solving a similar system of
equations to Section \ref{subsec:wTFExpansion}, but with an added unknown for the value of the wen, as well as added equations arising from
the wen relations.
Let $\glos{W} \in \calA^{sw}(\uparrow_1)$ denote the $Z$-value of the wen, and let us agree that the arrow diagram $W$ always appears just above the
wen on the skeleton. In fact, we are going to show that $W=1$ for any homomorphic expansion.
As two consecutive wens on the skeleton cancel, we obtain the equation shown in the picture and explained below:
\begin{center}
\input{figs/WenSquare.pstex_t}
\end{center}
The $Z$-value of two consecutive wens on a strand is a skeleton wen followed by $W$ followed by a skeleton wen and another $W$. Sliding the
bottom $W$ through the skeleton wen ``multiplies each tail by $(-1)$''. Let us denote this operation by ``bar'', i.e. for an arrow diagram $D$,
$\overline{D}=D\cdot(-1)^{\#\text{ of tails in }D}$. Cancelling the two skeleton wens, we obtain $\overline{W}W=1$. Recall that $\calA^{sw}(\up_1)$
consists only of wheels and single arrows. Since we are looking for a group-like $Z$, we can assume that $W=e^w$ and $\overline{W}W=1$ means
that $w$ is a linear combination of odd wheels and possibly single arrows.
Now recall the Twisted Vertex relation of Section \ref{subsubsec:WenRels}.
Note that the $Z$-value of the negative vertex on the right hand side of the relation can be
written as $S_1S_2A_1A_2(V)=\overline V$
(cf Remark~\ref{rem:SwitchingVertices}).
On the other hand, applying $Z$ to the left hand side of the relation we obtain:
\begin{center}
\input figs/VbarEquation.pstex_t
\end{center}
Hence, using that $\overline{W}=W^{-1}$, the twisted vertex relation induces the equation $W^1W^2=W^{(12)}$ in $\calA^{sw}(\up_2)$. One can verify degree by degree,
using that $W$ can be written as an exponential, that the only solution to this equation is $W=1$.
We have yet to verify that the CW relation (i.e., a cap can slide through a wen) can be satisfied with $W=1$.
Keep in mind that the wen as a skeleton feature anti-commutes
with tails (this is the Flip Relation of Section \ref{subsubsec:wTFGradRels}). The value of the cap $C$ is a combination of only wheels (Lemma \ref{lem:CapIsWheels}),
hence the CW relation translates to the equation $\overline{C}=C$, which is equivalent to saying that $C$ consists only of {\emph even} wheels.
The fact that $Z$ can be chosen to have this property
follows from Proposition 6.2 of \cite{AlekseevTorossian:KashiwaraVergne}: the value of the cap is $C=e^c$, where $c$ can be set to $c=-\frac{a}{2}$, as
explained in the proof of Theorem \ref{thm:ATEquivalence}. Here $a$ is such that $\tilde{\delta}(a)=jF$ as in Equation \eqref{eq:ATKVEqns}. A power series $f$
so that $a=\tr f$ (where $\tr$ is the trace which turns words into cyclic words) is called the Duflo function of $F$. In Proposition 6.2 Alekseev and Torossian
show that the even part of $f$ is $\frac{1}{2}\frac{\log(e^{x/2}-e^{-x/2})}{x}$, and that for any $f$ with this even part there is a corresponding solution $F$
of the generalized $KV$ problem. In particular, $f$ can be assumed to be even, namely the power series above, and hence it can be guaranteed that $C$ consists
of even wheels only. Thus we have proven the following:
\begin{theorem}\label{thm:WenATEquivalence}
Group-like homomorphic expansions $Z: \wTF \to \calA^{sw}$ (with no local arrows in the value of $V$) are in one-to one correspondence
with solutions to the KV problem with an even Duflo function. \qed
\end{theorem}
\draftcut
\subsection{Interlude: $u$-Knotted Trivalent Graphs}
\label{subsec:KTG}
The ``$u$sual'', or classical knot-theoretical objects corresponding to
$\wTF$ are loosely speaking Knotted Trivalent Graphs, or \glost{KTGs}.
We give a brief introduction/review of this structure before studying the
relationship between their homomorphic expansions and homomorphic expansions for
$\wTF$. The last goal of this paper is to show that the topological relationship between the
two spaces explains the relationship between the KV problem and Drinfel'd associators.
A trivalent graph is a graph with three edges meeting at each vertex,
equipped with a cyclic orientation of the three half-edges at each
vertex. KTGs are framed embeddings of trivalent graphs into $\bbR^3$,
regarded up to isotopies. The skeleton of a KTG is the trivalent
graph (as a combinatorial object) behind it. For a detailed
introduction to KTGs see for example \cite{Bar-NatanDancso:KTG}.
Here we only recall the most important facts. The reader might
recall that in Section~\ref{1-sec:w-knots}, the w-knot section,
of \cite{Bar-NatanDancso:WKO1} we only dealt with long $w$-knots,
as the $w$-theory of round knots is essentially trivial (see
\cite[Theorem~\ref{1-prop:AwCirc}]{Bar-NatanDancso:WKO1}). A similar issue
arises with ``$w$-knotted trivalent graphs''. Hence, the space we are
really interested in is ``long KTGs'', meaning, trivalent tangles with
1 or 2 ends.
\parpic[r]{\input{figs/UnzipAndInsertion.pstex_t}}
Long KTGs form an algebraic structure with operations as follows. {\em Orientation
switch} reverses the orientation of a specified edge. {\em Edge unzip} doubles a
specified edge as shown on the right. {\em Tangle
insertion} is inserting a small copy of a $(1,1)$-tangle $S$ into
the middle of some specified strand of a tangle $T$, as shown in the second row on
the right (tangle composition is a special case of this). The {\em stick-on} operation ``sticks a 1-tangle $S$ onto a specified edge of another
tangle $T$'', as shown. (In the figures $T$ is a 2-tangle, but this is irrelevant.) {\em Disjoint union} of
two 1-tangles produces a 2-tangle.
Insertion, disjoint union and stick-on are a slightly weaker set of operations than the connected sum
of~\cite{Bar-NatanDancso:KTG}.
The associated graded structure of the algebraic structure of long KTGs is the
graded space $\glos{\calA^u}$ of chord diagrams on
trivalent graph skeleta, modulo the $\glos{4T}$ and vertex invariance
(VI) relations. The induced operations on $\calA^u$ are as expected:
orientation switch multiplies a chord diagram by $(-1)$ to the number
of chord endings on the edge. The edge unzip $u_e$ maps a chord diagram
with $k$ chord endings on the edge $e$ to a sum of $2^k$ diagrams where
each chord ending has a choice between the two daughter edges. Finally,
tangle insertion, stick-on and disjoint union
induces the insertion, sticking on and disjoint union
of chord diagrams, respectively.
\parpic[r]{\input{figs/glitch.pstex_t}}
In \cite{Bar-NatanDancso:KTG} the authors prove that there is no
\emph{homomorphic} expansion for KTGs. This theorem, as well as the proof,
applies to long KTGs with slight modifications. However there are well-known --- and nearly homomorphic ---
expansions constructed by extending the Kontsevich integral to KTGs,
or from Drinfel'd associators. There are several such constructions
(\cite{MurakamiOhtsuki:KTGs}, \cite{ChepteaLe:EvenAssociator},
\cite{Dancso:KIforKTG}). For now, let us denote any one of these expansions by
$Z^{old}$. All $Z^{old}$ are ``almost homomorphic'': they intertwine every operation
except for edge unzip with their
chord-diagrammatic counterparts; but commutativity with unzip fails by
a controlled amount, as shown on the right. Here $\glos{\nu}$ denotes
the ``invariant of the unknot'', the value of which was conjectured in
\cite{Bar-NatanGaroufalidisRozanskyThurston:WheelsWheeling} and proven
in \cite{Bar-NatanLeThurston:TwoApplications}.
In \cite{Bar-NatanDancso:KTG} the authors fix this anomaly by slightly
changing the space of KTGs and adding some extra combinatorics (``dots''
on the edges), and construct a homomorphic expansion for this new space by
a slight adjustment of $Z^{old}$. Here we are going to use a similar but
different adjustment of the space of trivalent 1- and 2-tangles. Namely
we break the symmetry of the vertices and restrict the set of allowed
unzips.
\begin{definition}\label{def:sKTG} A ``signed KTG'', denoted $\glos{\sKTG}$, is
a trivalent oriented 1- or 2-tangle embedded in $\bbR^3$ with a cyclic orientation of edges meeting at each vertex,
and in addition each vertex is equipped with a sign and one of the three incident edges is marked as distinguished (sometimes denoted
by a thicker line). Our pictorial convention will
be that a vertex drawn in a ``\inverted{$Y$}'' shape
with all strands oriented up and the top strand distinguished is always
positive and a vertex drawn in a ``$Y$'' shape with strands oriented
up and the bottom strand distinguished is always negative (see Figure
\ref{fig:ZatVertices}).
\parpic[r]{\input{figs/StickOnSigns.pstex_t}}
The algebraic structure $\sKTG$ has one kind of objects for each skeleton (a skeleton is a uni-trivalent graph with signed vertices but
no embedding), as well as several operations: orientation switch, edge unzip, tangle insertion, disjoint union of 1-tangles, and stick-on.
Orientation switch of either of the non-distinguished strands changes the sign of the vertex, switching the orientation of
the distinguished strand does not. Unzip of an edge
is only allowed if the edge is distinguished at both of its ends and the vertices at either end are of opposite signs.
The stick-on operation can be done in either one of the two ways shown on the right (i.e., the stuck-on edge can be attached at
a vertex of either sign, but it can not become the distinguished edge of that vertex).
\end{definition}
To consider expansions of $\sKTG$, and ultimately the compatibility of
these with $Z^w$, we first note that $\sKTG$ is finitely generated (and
therefore any expansion $Z^u$ is determined by its values on finitely
many generators). The proof of this is not hard but somewhat lengthy,
so we postpone it to Section~\ref{subsec:sKTGgensProof}.
\begin{proposition}\label{prop:sKTGgens}
The algebraic structure $\sKTG$ is finitely generated by the following
list of elements:
\begin{center} \input{figs/sKTGgens.pstex_t} \end{center}
\end{proposition}
Note that we ignore strand orientations for simplicity in the statement of this proposition; this is not a problem as orientation switches are
allowed in $\sKTG$ without restriction.
\subsubsection{Homomorphic expansions for $\sKTG$}\label{subsubsec:Zu}
Suppose that $Z^u:\sKTG \to \calA^u$ is a homomorphic expansion. We hope to determine the value of $Z^u$ on each of the generators.
\parpic[r]{\input{figs/BubbleSquared.pstex_t}}
The value of $Z^u$ on the single strand is an element of $\calA^u(\uparrow)$ whose square is itself, hence it is 1.
The value of
the bubble is an element $x \in \calA^u(\uparrow_2)$, as all chords can be pushed to the ``bubble'' part using the VI relation. Two bubbles
can be composed and unzipped to produce a single bubble (see on the right), hence we have $x^2=x$, which implies $x=1$
in $\calA^u(\uparrow_2)$.
Recall that a Drinfel'd associator is a group-like
element $\Phi \in \calA^u(\uparrow_3)$ along with a group-like element $R^u \in \calA^u(\up_2)$ satisfying the so-called pentagon and
positive and negative hexagon equations, as well as a non-degeneracy
and mirror skew-symmetry property. For a detailed explanation see
Section 4 of \cite{Bar-NatanDancso:KTG}; associators were first defined
in \cite{Drinfeld:QuasiHopf}. We claim that the $Z^u$-value $\glos{\Phi}$ of the right
associator, along with the value $\glos{R^u}$ of the right twist forms a Drinfel'd associator pair. The proof of this statement
is the same as the proof of Theorem 4.2 of \cite{Bar-NatanDancso:KTG},
with minor modifications (making heavy use of the assumption
that $Z^u$ is homomorphic).
It is easy to check by composition and unzips that the value of the
left associator and the left twist are $\Phi^{-1}$ and $(R^u)^{-1}$, respectively.
Note that if $\Phi$ is a {\em horizontal chord} associator (i.e.,
all the chords of $\Phi$ are horizontal on three strands) then $R^u$ is forced to
be $e^{c/2}$ where $c$ denotes a single chord. Note that the reverse is not true:
there exist non-horizontal chord associators $\Phi$ that satisfy the hexagon equations with $R^u=e^{c/2}$.
\parpic[r]{\input{figs/NooseBalloonPhi.pstex_t}}
Let $b$ and $n$ denote the $Z^u$-values of the balloon and the noose, respectively. Note that using the $VI$ relation all chord endings
can be pushed to the ``looped'' strands, so $b$ and $n$ live in $\calA^u(\uparrow)$, as seen in Figure \ref{fig:NooseBalloonProof}. The argument in that figure
shows that $n\cdot b$ is the inverse in $\calA^u(\uparrow)$ of ``an associator on a squiggly strand'', as shown on the right.
In Figure \ref{fig:NooseBalloonProof} we start with the $\sKTG$ on the top left and either apply $Z^u$ followed by unzipping the
edges marked
by stars, or first unzip the same edges and then apply $Z^u$.
Since $Z^u$ is homomorphic, the two results in the bottom right corner must agree.
(Note that two of the four unzips we perform are ``illegal'',
as the strand directions can't match. However, it is easy to get around this issue by inserting small bubbles at the top of the balloon and the bottom
of the noose, and switching the appropriate edge orientations before and after the unzips. The $Z^u$-value of a bubble is 1, hence this will not effect
the computation and so we ignore the issue for simplicity.)
\begin{figure}[h]
\input{figs/NooseBalloonProof.pstex_t}
\caption{Unzipping a noose and a balloon to a squiggle.}
\label{fig:NooseBalloonProof}
\end{figure}
In addition, it follows from Theorem 4.2 of \cite{Bar-NatanDancso:KTG} via deleting two edges
that the inverse of an ``associator on a squiggly strand'' is $\nu$, the invariant of the unknot.
To summarize, we have proven the following:
\begin{lemma}\label{lem:nb} If $Z^u$ is a homomorphic expansion then the $Z^u$ values of
the strand and the bubble are 1, the values of the right associator and right twist form an associator pair $(\Phi,R^u)$,
and the values of the left twist and left associator are inverses of these.
With $n$ and $b$ denoting the value of the noose and the balloon, respectively, and $\nu$
being the invariant of the unknot, we have $n \cdot b =\nu$ in $\calA^u(\uparrow)$.
\end{lemma}
The natural question to ask is whether any triple $(\Phi, R^u, n)$ gives rise to a homomorphic expansion. We don't know
whether this is true, but we do know that any pair $(\Phi, R^u)$ gives rise to a ``nearly homomorphic'' expansion of KTGs
\cite{MurakamiOhtsuki:KTGs, ChepteaLe:EvenAssociator, Dancso:KIforKTG},
and we can construct a homomorphic expansion for $\sKTG$ from any of these (as shown below). However, all of these expansions
take the same specific value on the noose and the balloon (also see below). We don't know whether there really is a one
parameter family of homomorphic expansions $Z^u$ for each choice of $(\Phi, R^u)$ or if we are simply missing
a relation.
\parpic[r]{\input{figs/ZoldOfTangle.pstex_t}}
We now construct explicit homomorphic expansions $Z^u \colon \sKTG \to \calA^u$ from any $Z^{old}$ (where $Z^{old}$ stands for
an ``almost homomorphic'' expansion of KTGs) as follows. First of all we need to interpret
$Z^{old}$ as an invariant of 2-tangles. This can be done by connecting the top and bottom ends by a non-interacting long
strand followed by a normalization, as shown on the right. By ``multiplying by $\nu^{-1}$'' we mean that after computing $Z^{old}$
we insert $\nu^{-1}$ on the long strand (recall that $\nu$ is the ``invariant of the unknot''). We interpret $Z^{old}$ of a 1-tangle
as follows: stick the 1-tangle onto a single strand to obtain a 2-tangle, then proceed as above. The result will only have chords on the
1-tangle (using that the extensions of the Kontsevich Integral are homomorphic with respect to ``connected sums''),
so we define the result to be the value of $Z^{old}$ on the 1-tangle.
As an example, we compute the value of $Z^{old}$ for the noose in Figure \ref{fig:uValueNoose}
(note that the computation for the balloon is the same).
\begin{figure}
\input{figs/uValueOfTheNoose.pstex_t}
\caption{Computing the $Z^{old}$ value of the noose. The third step uses that the Kontsevich integral of KTGs is homomorphic with
respect to the ``connected sum'' operation and that the value of the unknot is $\nu$ (see \cite{Bar-NatanDancso:KTG} for an explanation of
both of these facts).}
\label{fig:uValueNoose}
\end{figure}
\begin{figure}[h]
\input{figs/ZatVertices.pstex_t}
\caption{Normalizations for $Z^u$ at the vertices.}\label{fig:ZatVertices}
\end{figure}
\parpic[r]{\input{figs/NooseBalloonValue.pstex_t}}
Now to construct a homomorphic $Z^u$ from $Z^{old}$ we add normalizations
near the vertices,
as in Figure~\ref{fig:ZatVertices}, where $c$ denotes a single chord.
Checking that $Z^u$ is a homomorphic expansion is a simple calculation
using the almost homomorphicity of $Z^{old}$, which we leave to the
reader. The reader can also verify that $Z^u$ of the strand and the bubble
is 1 as it should be. $Z^u$ of the right twist is $e^{c/2}$ and $Z^u$ of the right associator is a Drinfel'd associator
$\Phi$ (note that $\Phi$ depends on which $Z^{old}$ was used).
From the calculation of Figure \ref{fig:uValueNoose} it follows that the $Z^u$
value of the balloon and the noose (for any $Z^{old}$) are as shown on the right,
and indeed $n\cdot b = \nu$.
\subsection{The relationship between $\sKTG$ and $\wTF$}\label{subsec:wTFcompatibility}
We move on to the question of compatibility between the homomorphic expansions
$Z^u$ and $Z^w$ (from now on we are going to refer to the homomorphic
expansion of $\wTF$ --- called $Z$ in the previous section --- as $Z^w$
to avoid confusion).
There is a map $a\colon \sKTG \to \wTF$, given by interpreting $\sKTG$
diagrams as $\wTF$ diagrams. In particular, positive vertices (of edge
orientations as shown in Figure \ref{fig:ZatVertices}) are interpreted as the positive $\wTF$ vertex
\input{figs/PlusVertex.pstex_t} and negative vertices as the negative vertex
\input{figs/NegVertex.pstex_t}. (The map $a$ can also be interpreted topologically as
Satoh's tubing map.) The induced map $\alpha\colon \calA^u
\to \calA^{sw}$ is as defined in Section \ref{subsec:sder}, that is,
$\alpha$ maps each chord to the sum of its two possible orientations.
Hence we can ask whether the two expansions are compatible (or can
be chosen to be compatible), which takes us to the main result of
this section:
\def\uwsquare{{\xymatrix{
\sKTG \ar[d]^{Z^u} \ar[r]^a & \wTF \ar[d]^{Z^w} \\
\calA^u \ar[r]^\alpha & \calA^{sw}
}}}
\parpic(2in,1in)[r]{\null\raisebox{-5mm}{\begin{minipage}{2in}
\begin{equation} \label{eq:uwcompatibility}
\begin{array}{c} \uwsquare \end{array}
\end{equation}
\end{minipage}}}
\begin{theorem}\label{thm:ZuwCompatible}
\picskip{4}
Let $Z^u$ be a homomorphic expansion for $\sKTG$ with the properties that $\Phi$
is a horizontal chord associator and $n=e^{-c/4}\nu^{1/2}$ in the sense of Section \ref{subsubsec:Zu}.\footnote{It will become apparent that in the
proof we only use slightly weaker but less aesthetic conditions on $Z^u$.}
Then there exists a homomorphic expansion $Z^w$ for $\wTF$ compatible with $Z^u$ in the sense
that the square on the right commutes.
Furthermore, such $Z^w$ are in one to one correspondence\footnote{An even nicer theorem would be a classification of
homomorphic expansions for the combined algebraic structure $\left(\sKTG\overset{a}{\longrightarrow}\wTF\right)$ in terms of solutions of
the KV problem. The two obstacles to this are clarifying whether there is a free choice of $n$ for $Z^u$, and --- probably much harder --- how much of the
horizontal chord condition is necessary for a compatible $Z^w$ to exist.} with
``symmetric solutions of the KV problem'' satisfying the KV equations \eqref{eq:ATKVEqns}, the ``twist equation''
\eqref{eq:TwistWithF} and the associator equation \eqref{eq:ATPhiandV}.
\end{theorem}
\picskip{0}Before moving on to the proof let us state and prove the following Lemma,
to be used repeatedly in the proof of the theorem.
\begin{lemma}\label{lem:TreesAndUnitarity}
If $a$ and $b$ are group-like elements in $\calA^{sw}(\uparrow_n)$, then $a=b$ if and only if $\pi(a)=\pi(b)$ and $aa^*=bb^*$. Here $\pi$
is the projection induced by $\pi\colon \calP^w(\uparrow_n) \to \tder_n \oplus \fraka_n$ (see Section \ref{subsec:ATSpaces}),
and $*$ refers to the adjoint map of Definition \ref{def:Adjoint}. In the notation of this section $*$ is applying the adjoint $A$ on all strands.
\end{lemma}
\begin{proof}
Write $a=e^we^{uD}$ and $b=e^{w'}e^{uD'}$, where $w\in \attr_n$, $D\in \tder_n\oplus \fraka_n$
and $u\colon \tder_n\oplus \fraka_n \to \calP_n$ is the ``upper'' map of
Section \ref{subsec:ATSpaces}. Assume that $\pi(a)=\pi(b)$ and $aa^*=bb^*$. Since $\pi(a)=e^D$ and $\pi(b)=e^{D'}$, we conclude
that $D=D'$. Now we compute $aa^*=e^we^{uD}e^{-lD}e^w=e^we^{j(D)}e^w,$ where $j\colon \tder_n \to \attr_n$ is the map defined in Section 5.1 of
\cite{AlekseevTorossian:KashiwaraVergne} and discussed in \ref{prop:Jandj} of this paper. Now note that both $w$ and $j(D)$ are elements of
$\attr_n$, hence they commute, so $aa^*=e^{2w+j(D)}$. Thus, $aa^*=bb^*$ means that $e^{2w+j(D)}=e^{2w'+j(D)}$, which implies that $w=w'$ and
$a=b$. \qed
\end{proof}
\noindent{\em Proof of Theorem \ref{thm:ZuwCompatible}.}
In addition to being a homomorphic expansion for $\wTF$, $Z^w$ has to satisfy an
the added condition of being compatible with $Z^u$. Since $\sKTG$ is finitely generated, this translates
to one additional equation for each generator of $\sKTG$, some of which are
automatically satisfied. To deal with the others, we use the machinery established in the previous sections
to translate these equations to conditions on $F$, and they turn out to be the properties studied in \cite{AlekseevTorossian:KashiwaraVergne}
which link solutions of the KV problem with Drinfel'd associators.
To start, note that for the single strand and the bubble the commutativity of the square \eqref{eq:uwcompatibility} is
satisfied with any $Z^w$: both the $Z^u$ and $Z^w$ values are 1 (note that the $Z^w$ value of the bubble
is 1 due to the unitarity \eqref{eq:unitarity} of $Z^w$). Each of the other generators will require more study.
{\em Commutativity of~\eqref{eq:uwcompatibility}
for the twists.} Recall that the $Z^u$-value of the right twist (for a $Z^u$ with horizontal chord $\Phi$)
is $R^u=e^{c/2}$; and note that its $Z^w$-value is $V^{-1}RV^{21}$,
where $R=e^{a_{12}}$ is the $Z^w$-value of the crossing (and $a_{12}$ is a single arrow pointing from strand 1 to strand 2). Hence the
commutativity of \eqref{eq:uwcompatibility} for the right twist is equivalent to the
``Twist Equation'' $\alpha(R^u)=V^{-1}RV^{21}$.
By definition
of $\alpha$, $\alpha(R^u)=e^{\frac{1}{2}(a_{12}+a_{21})}$, where $a_{12}$ and $a_{21}$ are single arrows pointing from strand 1 to 2 and 2 to 1,
respectively. Hence we have
\begin{equation}\label{eq:twist}
e^{\frac{1}{2}(a_{12}+a_{21})}=V^{-1}RV^{21}.
\end{equation}
To translate this to the language of \cite{AlekseevTorossian:KashiwaraVergne}, we use
Lemma \ref{lem:TreesAndUnitarity}, which implies that it is enough
for $V$ to satisfy the Twist Equation ``on tree level'' (i.e., after applying $\pi$), and for which the adjoint condition of the Lemma holds.
We first prove that the adjoint condition holds for any homomorphic expansion of $\wTF$. Multiplying the left hand side of the Twist Equation by its adjoint, we get
$$e^{\frac{1}{2}(a_{12}+a_{21})}(e^{\frac{1}{2}(a_{12}+a_{21})})^*=e^{\frac{1}{2}(a_{12}+a_{21})}e^{-\frac{1}{2}(a_{12}+a_{21})}=1.$$
As for the right hand side, we have to compute $V^{-1}RV^{21}(V^{21})^*R^*(V^{-1})^*$. Since $V$ is unitary (Equation (\ref{eq:unitarity})), $VV^*=V\cdot A_1A_2(V)=1$.
Now $R=e^{a_{12}}$, so $R^*=e^{-a_{12}}=R^{-1}$, hence the expression on the right hand side also simplifies to 1, as needed.
As for the ``tree level'' of the Twist Equation, recall that in Section \ref{subsec:wTFExpansion} we used
Alekseev and Torossian's solution $F\in \TAut_2$ to the Kashiwara--Vergne
equations \cite{AlekseevTorossian:KashiwaraVergne} to find solutions $V$ to equations (\ref{eq:HardR4}),(\ref{eq:unitarity}) and (\ref{eq:CapEqn}).
We produced $V$ from $F$ by setting $F=e^{D^{21}}$ with $D\in \tder_2^s$, $b:=\frac{-j(F)}{2}\in \attr_2$ and $V:=e^be^{uD}$, so $F$ is ``the tree part'' of $V$,
up to re-numbering strands. Hence, the tree level Twist Equation translates to a new equation for $F$.
Substituting $V=e^be^{uD}$ into the Twist Equation we obtain
$ e^{\frac{1}{2}(a_{12}+a_{21})}=e^{-uD}e^{-b}e^{a_{12}}e^{b^{21}}e^{uD^{21}},$
and applying $\pi$, we get
\begin{equation}\label{eq:TwistWithF}
e^{\frac{1}{2}(a_{12}+a_{21})}=(F^{21})^{-1}e^{a_{12}}F.
\end{equation}
In \cite{AlekseevTorossian:KashiwaraVergne} the solutions $F$ of the KV equations which also satisfy
this equation are called ``symmetric solutions of
the Kashiwara-Vergne problem'' discussed in Sections 8.2
and 8.3. (Note that in
\cite{AlekseevTorossian:KashiwaraVergne} $R$ denotes $e^{a_{21}}$).
{\em Commutativity of~\eqref{eq:uwcompatibility} for the
associators.}
Recall that the $Z^u$ value of the right associator is a Drinfel'd associator $\Phi \in \calA^u(\uparrow_3)$;
for the $Z^w$ value see Figure \ref{fig:wAssociator}.
Hence the new condition on $V$ is the following:
\begin{equation}\label{eq:AssociatorAndV}
\alpha(\Phi)=V_-^{(12)3}V_-^{12}V^{23}V^{1(23)}
\qquad\text{in}\qquad
\calA^{sw}(\uparrow_3)
\end{equation}
\begin{figure}
\input{figs/Associator.pstex_t}
\caption{The $Z^w$-value of the right associator.}
\label{fig:wAssociator}
\end{figure}
Again we treat the ``tree and wheel parts'' separately
using Lemma \ref{lem:TreesAndUnitarity}. As $\Phi$ is by
definition group-like, let us denote $\Phi=:\glos{e^\phi}$.
We first verify that the ``wheel part'' or adjoint condition of the Lemma holds. Starting
with the right hand side of Equation~\eqref{eq:AssociatorAndV}, the
unitarity $VV^*=1$
of $V$ implies that
\[ V_-^{(12)3}V_-^{12}V^{23}V^{1(23)}
(V^{1(23)})^*(V^{23})^*(V_-^{12})^*(V_-^{(12)3})^*=1.
\]
For the left hand side of~\eqref{eq:AssociatorAndV} we need to show that
$e^{\alpha(\phi)}(e^{\alpha(\phi)})^*=1$ as well, and this is true for
any {\em horizontal chord} associator.
Indeed, restricted to the $\alpha$-images of horizontal
chords $*$ is multiplication by $-1$, and as it is an anti-Lie morphism,
this fact extends to the Lie algebra generated by $\alpha$-images
of horizontal chords. Hence $e^{\alpha(\phi)}(e^{\alpha(\phi)})^*
=e^{\alpha(\phi)}e^{\alpha(\phi)^*}=e^{\alpha(\phi)}e^{-\alpha(\phi)}=1$.
On to the tree part, applying $\pi$ to Equation (\ref{eq:AssociatorAndV})
and keeping in mind that $V_-=V^{-1}$ by the unitarity of $V$, we obtain
\begin{multline}\label{eq:ATPhiandV}
e^{\pi\alpha(\phi)}
= (F^{3(12)})^{-1} (F^{21})^{-1} F^{32} F^{(23)1}
=e^{-D^{(12)3}}e^{-D^{12}}e^{D^{23}}e^{D^{1(23)}} \\
\text{in }
\glos{\SAut_3}:=\exp(\sder_3)\subset\TAut_3.
\end{multline}
This is Equation (26) of \cite{AlekseevTorossian:KashiwaraVergne},
up to re-numbering strands 1 and 2 as 2 and 1\footnote{Note that
in \cite{AlekseevTorossian:KashiwaraVergne} ``$\Phi'$ is an
associator'' means that $\Phi'$ satisfies the pentagon equation,
mirror skew-symmetry, and positive and negative hexagon equations
in the space $\SAut_3$. These equations are stated in
\cite{AlekseevTorossian:KashiwaraVergne} as equations (25), (29),
(30), and (31), and the hexagon equations are stated with strands 1
and 2 re-named to 2 and 1 as compared to \cite{Drinfeld:QuasiHopf}
and \cite{Bar-NatanDancso:KTG}. This is consistent with $F=e^{D^{21}}$.}.
The following fact from
\cite{AlekseevTorossian:KashiwaraVergne} (their Theorem 7.5, Propositions
9.2 and 9.3 combined) implies that there is a solution $F$ to the KV equations \eqref{eq:ATKVEqns}
which also satisfies \eqref{eq:TwistWithF} and \eqref{eq:ATPhiandV}.
\begin{fact}
If $\Phi'=e^{\phi'}$ is an associator in $\SAut_3$ so that $j(\Phi')=0$\footnote{The condition
$j(\phi')=0$ is equivalent to the condition $\Phi\in KRV^0_3$ in \cite{AlekseevTorossian:KashiwaraVergne}.
The relevant definitions in \cite{AlekseevTorossian:KashiwaraVergne} can be found in Remark 4.2 and at the bottom of
page 434 (before Section 5.2).}
then Equation~(\ref{eq:ATPhiandV}) has a solution $F=e^{D^{21}}$ which is
also a solution to the KV equations, and all such solutions are symmetric
(i.e. verify the Twist Equation (\ref{eq:TwistWithF})). \qed
\end{fact}
To use this Fact, we need to show that $\Phi':=\pi\alpha(\Phi)$ is an
associator in $\SAut_3$ and that $j(\Phi')=j(\pi\alpha(\Phi))=0$. The
latter is the unitarity of $\Phi$ which is already proven. The
former follows from the fact that $\Phi$ is an associator and the fact
(Theorem~\ref{thm:sder}) that the image of $\pi\alpha$ is contained in
$\sder$ (ignoring degree 1 terms, which are not present in an associator
anyway).
In summary, the condition of the Fact are satisfied and so there exists a solution $F$ which
in turn induces a $Z^w$ which is compatible with $Z^u$ for the strand, the bubble, the twists and
the associators. That is, all
generators of $\sKTG$ except possibly the balloon and the noose. As the last step of the proof
of Theorem \ref{thm:ZuwCompatible} we show that any such $Z^w$ also automatically make
\eqref{eq:uwcompatibility} commutative for the balloon and the noose.
\parpic[r]{\input{figs/wValueOfTheNoose.pstex_t}}
{\em Commutativity of~\eqref{eq:uwcompatibility}
for the balloon and the noose.} Since we know the $Z^u$-values $B$ and $n$ of the balloon and the noose,
we start by computing $Z^w$ of the noose. $Z^w$ assigns a $V$ value to the vertex with the
first strand orientation switched as shown in the figure on the
right. The balloon is the same, except with a negative vertex and the second strand reversed.
Hence what we need to show is that the two equations
below hold:
\begin{center}
\input{figs/NooseEquations.pstex_t}
\end{center}
Let us denote the left hand side of the first equation above by $n^w$ and $b^w$ (the $Z^w$ value of the noose
and the balloon, respectively).
We will start by proving that the product of these two equations holds,
namely that $n^wb^w=\alpha(\nu)$.
(We used that any local (small) arrow diagram on a single strand is central in
$\calA^{sw}(\uparrow_n)$, hence the cancellations.)
This product equation is satisfied due to an argument identical to that of
Figure~\ref{fig:NooseBalloonProof}, but carried out in $\wTF$, and using
that by the compatibility with associators, $Z^w$ of an associator is $\alpha(\Phi)$.
What remains is to show that the noose and balloon equations hold individually. In light of the results so far, it is sufficient to show that
\begin{equation}\label{eq:NooseSymmetry}
n^w=b^w\cdot e^{-D_A},
\end{equation}
where $D_A$ stands for a single arrow on one strand (whose direction
doesn't matter due to the $RI$ relation. As stated in
\cite[Theorem~\ref{1-thm:Aw}]{Bar-NatanDancso:WKO1},
$\calA^{sw}(\uparrow_1)$ is the polynomial algebra freely generated by the
arrow $D_A$ and wheels of degrees 2 and higher. Since $V$ is group-like,
$n^w$ (resp. $b^w$) is an exponential $e^{A_1}$ (resp. $e^{A_2}$)
with $A_1, A_2 \in \calA^{sw}(\uparrow_1)$. We want to show that
$e^{A_1}=e^{A_2}\cdot e^{-D_A}$, equivalently that $A_1=A_2-D_A$.
\begin{figure}
\input figs/NooseCappedProof.pstex_t
\caption{The proof of Equation (\ref{eq:NooseCapped}). Note that the unzips are ``illegal'', as the strand directions don't match. This can be fixed
by inserting a small bubble at the bottom of the noose and doing a number of orientation switches. As this doesn't change the result or the main argument,
we suppress the issue for simplicity. Equation (\ref{eq:NooseCapped}) is obtained from this result by multiplying by $S(C)^{-1}$ on the bottom and by $C^{-1}$
on the top.}\label{fig:NooseCappedProof}
\end{figure}
In degree 1, this can be done by explicit verification. Let $A_1^{\geq
2}$ and $A_2^{\geq 2}$ denote the degree 2 and higher parts of $A_1$ and
$A_2$, respectively. We claim that capping the strand at both its top
and its bottom takes $e^{A_1}$ to $e^{A_1^{\geq 2}}$, and similarly $e^{A_2}$
to $e^{A_2^{\geq 2}}$. (In other words, capping kills arrows but leaves
wheels un-changed.) This can be proven similarly to the proof of
Lemma~\ref{lem:CapIsWheels}, but using
\[
F' := \sum_{k_1,k_2=0}^{\infty}
\frac{(-1)^{k_1+k_2}}{k_1!k_2!}D_A^{k_1+k_2}S_L^{k_1}S_R^{k_2}
\]
in place of $F$ in the proof. What we need to prove, then, is the following equality, and
the proof is shown in Figure~\ref{fig:NooseCappedProof}.
\begin{equation}\label{eq:NooseCapped}
\raisebox{-6mm}{\input figs/NooseCapped.pstex_t}
\end{equation}
This concludes the proof of Theorem~\ref{thm:ZuwCompatible}. \qed
\vspace{2mm}
Recall from Section~\ref{subsec:sder} that there is no commutative
square linking $Z^u\colon\uT\to\calA^u$ and $Z^w\colon\wT\to\calA^{sw}$,
for the simple reason that the Kontsevich integral for tangles $Z^u$
is not canonical, but depends on a choice of parenthesizations for
the ``bottom'' and the ``top'' strands of a tangle $T$. Yet given
such choices, a tangle $T$ can be ``closed up with trees'' as within the proof of
Proposition~\ref{prop:sKTGgens} (see Section \ref{sec:odds}) into an $\sKTG$ which we will denote
$G$. For $G$ a commutativity statement does hold as we have just
proven. The $Z^u$ and $Z^w$ invariants of $T$ and of $G$ differ only
by a number of vertex-normalizations and vertex-values on skeleton-trees
at the bottom or at the top of $G$, and using VI, these values can slide
so they are placed on the original skeleton of $T$. This is summarized
as the following proposition:
\begin{proposition} \label{prop:uwBT} Let $n$ and $n'$ be natural numbers.
Given choices $c$ and and $c'$ of parenthesizations of $n$ and $n'$
strands respectively, there exists invertible elements
$C\in\calA^{sw}(\uparrow_n)$ and $C'\in\calA^{sw}(\uparrow_{n'})$ so
that for any u-tangle $T$ with $n$ ``bottom'' ends and $n'$ ``top'' ends
we have
\[ \alpha Z^u_{c,c'}(T)=C^{-1}Z^w(aT)C', \]
where $Z^u_{c,c'}$ denotes the usual Kontsevich integral of $T$ with
bottom and top parenthesizations $c$ and $c'$.
\end{proposition}
For u-braids the above proposition may be stated with $c=c'$ and then $C$
and $C'$ are the same.