\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:wTF}
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. In this case, we use a {\em two-coloured} circuit algebra. We will see that in the local topological interpretation the two colours correspond to opposite surface orientations. To give a finite presentation for a circuit algebra with auxiliary (additional) operations, we use the notation
$$\CA\left\langle  \parbox{1.5in}{\centering Circuit algebra generators} \left| \parbox{1.5in}{\centering Circuit algebra relations} \right|  \parbox{1.2in}{\centering Auxiliary operations} \right\rangle.$$

\begin{definition}\label{def:wTF} Let $\glos{\wTF}$ denote the two-coloured circuit algebra -- where the colours are denoted by {\em thick black} and {\em hollow blue} strands -- given by the following generators, relations and auxiliary operations:
 \[
  \wTF=\CA\!\left\langle\left.\left.
  \raisebox{-6mm}{\input{figs/wTFgensWen@.pstex_t}}
  \right|
  \parbox{1.9in}{\centering \Rs, R2, R3, R4, OC, CP, FR, $W^2$, CW, TV}
  \right|
  \parbox{0.9in}{\centering $S_e, u_e, d_e, \mu$}
  \right\rangle.
\]

\end{definition}

\parpic[r]{\input{figs/VertexExamples.pstex_t}}
The generators consist of crossings, caps, wens, and foam vertices, each shown above in all permissible colour combinations. Note that the foam vertices, where three strands meet, also come in all possible combinations of strand directions, of which we only show two for brevity. Some additional examples are shown on the right. These generators are related by the orientation switch operation $S_e$, whose topological interpretation is explained in Section~\ref{subsubsec:wops}.

The relations \Rs, R2, R3 and $OC$ are as in Section~\ref{sec:w-tangles}. The other relations are shown and explained in the context of their local topological meaning in Section~\ref{subsubsec:wrels}. 

A {\em strand} of a w-tangled foam is a line between two vertices, tangle ends (boundary points), or caps; strands may go over and under multiple crossings. The auxiliary operations of $\wTF$ are strand orientation switches $S_e$, strand unzips $u_e$, deletions $d_e$ of {\em long strands} which end in two tangle ends,  and a global surface orientation reversal. These are described, along with their topological interpretations, in Section~\ref{subsubsec:wops}.

The circuit algebra $\wTF$ is skeleton-graded
where the circuit algebra of skeleta $\calS$ is a version of the skeleton algebra $\calS$ introduced
in Section~\ref{subsec:CircuitAlgebras}, but here two-coloured and with vertices and caps included:
 \[
  \calS=\CA\!\left.\left\langle
  \raisebox{-2mm}{\input{figs/SkelGenWen.pstex_t}}
  \right|
  \parbox{1.2in}{\centering $W^2$, CW, TV}
  \right\rangle.
\]
Denote by $\sigma: \wTF \to \calS$ the skeleton map, given by $\sigma(\overcrossing)=\sigma(\undercrossing)=\virtualcrossing$, for any (consistent) colouring of the crossings. All other generators are mapped to themselves.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

\begin{comment}\label{com:MissingTopology}
We conjecture that the generators and relations of $\wTF$ provide a Reidemeister theory for this topological interpretation of w-tangled foams. However, there is no complete Reidemeister theorem even for w-knots (see \cite[Section 3]{Bar-NatanDancso:WKO1}). For any rigorous purposes below,
$\wTF$ is studied as a circuit algebra given by generators and relations,
with topology serving only as intuition.
\end{comment}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{The generators of $\wTF$}\label{subsubsec:wTFgens}
There is topological meaning to each of the generators
of $\wTF$: via a generalzation of the Satoh tubing map
of Section~\ref{sec:w-tangles} they each stand for
local features of framed knotted ribbon tubes in $\bbR^4$.

\parpic[r]{\input{figs/ThickAndHollow.pstex_t}}
The map $\delta$ treats the thick black strands in the same way as in Section~\ref{sec:w-tangles}, and treats the
hollow blue strands also in the same way, except reversing the 2D orientation at the target.

The black crossings are as explained in Section
\ref{subsec:TangleTopology}: the under-strand denotes the small circle flying through a larger one,
or, equivalently, a ``thin'' tube braided through a thicker one. Recall that for tangles there are four kinds of crossings (left or right circle flying through from below or from above the other), but the two kinds that are not shown as CA generators are obtained from the generators by adding virtual crossings (see Figures~\ref{fig:CrossingTubes} and~\ref{fig:BandCrossings}). The blue and mixed-colour crossings in $\wTF$ represent the same braid but with different surface orientations. In particular, note that reversing the colour of the top strand reverses the fly-through direction, as shown in Figure~\ref{fig:ColouredCrossings}.
A black positive vertex represents a small circle (represented by the under strand) flying though a larger circle (represented by the under strand) for below.
If the over strand is hollow blue, then the corresponding tube has the opposite surface orientation, and so do the large circles of the corresponding movie. Hence the meaning of the words "above" and "below" is reversed. Thus, a positive vertex with blue over strand represents the small circle flying through the larger one from above.


\begin{figure}
\input{figs/ColouredCrossings.pstex_t}
\caption{Coloured crossings.}\label{fig:ColouredCrossings}
\end{figure}



\parpic[r]{\input{figs/Cap.pstex_t}}
A bulleted end -- in either colour -- denotes a cap on the tube, or a flying circle that shrinks to a point, 
as in the figure on the right.

\vskip 5mm

The $\glos{w}$ marking on a strand indicates a
{\em wen}. A wen is a Klein bottle cut apart (see
\cite[Section~\ref{1-subsubsec:NonHorRings}]{Bar-NatanDancso:WKO1});
in other words it changes the 2D orientation of a tube, as shown below in band and strand notations and surface pictures:
\begin{center}
 \input{figs/Wen2@.pstex_t}
\end{center}



\parpic[l]{\includegraphics[height=5cm]{figs/TheVertex.ps}}
The final generators -- in either colour -- denote {\em singular foam vertices}. As the notation
suggests, a vertex can be thought of as a crossing with either the bottom or the top half tubes identified. To make
this precise using the flying circles interpretation, the first black
vertex represents the movie shown on the left: the circle corresponding
to the right strand approaches the ring represented by the left strand 
from below, flies inside it, and then the two rings fuse (as opposed to
a crossing where the ring coming from the right would continue to fly
out to above and to the left of the other one): see the figure on the left. Note that the colours in this figure are illustrative only, and do not reference the black/blue circuit algebra colouring.   
The second black 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. As in the case of crossings, the fly-in direction is opposite for the hollow blue versions of the vertices.

The sign of a vertex is defined the same way as the sign of a crossing
(see Section~\ref{subsec:TangleTopology}).  The first black generator vertex is positive, and we often refer to it as ``{\em the} positive vertex'', although its blue counterpart is also positive. Similarly, we call  the second black generating vertex
``{\em the} negative vertex''.  

As with crossings, we obtain the vertices with opposite fly-in directions by composing the generating vertices with virtual crossings, as shown
in Figure~\ref{fig:VertexTypes}. In the figure the band notation for vertices is used
the same way as it is for crossings: the fully coloured band stands for
the thin (inner) ring. The colours in the figure are illustrative only, and do not reference the black/blue circuit algebra colouring.  


\begin{figure}[h!]
\input{figs/VertexTypes.pstex_t}
\caption{Vertex types in $\wTFo$.  }\label{fig:VertexTypes}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{The relations of $\wTF$} \label{subsubsec:wrels} 
Next, we discuss the relations of $\wTF$ and show that they represent local isotopies of w-foams.

The usual \Rs, R2, R3, and OC relations of
Figure~\ref{fig:VKnotRels} continue to apply, given all black strands, that is, blackboard orientation throughout. Basic Reidemeister relations for different orientations can be derived by combining the black Reidemeister relations with the Flip Relations (which govern the interactions of wens with crossings), see Remark~\ref{rem:BlueReidemeister}.
The additional relations are necessary to describe the interactions of the additional features.

The Reidemeister 4 (\glost{R4}) relations assert that a (black) strand can be moved under
or over a (black) vertex, as shown below. The ambiguously
drawn vertices in the figure denote a black vertex of any sign with any strand directions (as in Section~\ref{subsubsec:wTFgens}). The local isotopies can be read from the band pictures in the bottom row. Reidemeister 4 relations for the blue vertices can be derived from the black R4 relations and the various wen relations, see Remark~\ref{rem:BlueReidemeister}.
\begin{center}
 \input{figs/R4.pstex_t}
\end{center}



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. We
denote this relation by \glost{CP}, for Cap Pull-out.
\[ \input{figs/CapRel.pstex_t} \]
This is the case for any strands directions, and in fact in either colour (though the blue CP relation can also be obtained by combining the black CP relation with several wen relations). 

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

The interaction of a wen and a crossing is described by the following
Flip Relations (\glost{FR}):
\begin{center}
 \input figs/FlipRels2@.pstex_t
\end{center}
To explain the isotopy (movie homotopy) represented by this relation, note that in the flying circle interpretation, 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, hence the virtual crossings 

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

\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, as illustrated on the left. The two lower strands of the vertex represent two merging circles. In the band notation there is a non-filled and a filled band (corresponding to large circle/ small circle in the movie interpretation). There 
is also a top and a bottom band: 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 denote this relation by
\glost{TV}, for {\em Twisted Vertex}. Translated to diagrams, the relations shown below are imposed in $\wTF$:

\begin{center}
\input figs/TVRel.pstex_t
\end{center}

\begin{remark}\label{rem:BlueReidemeister}
Versions of the Reidemeister relations for different surface orientations can be derived by combining the relations for the default (black) orientation with the wen relations. For example, the blue R2 relation is shown to be the same as its black counterpart on the left side of Figure~\ref{fig:BlueRelations}
Following similar reasoning, however, we obtain that the R4 relation for moving a strand under a blue vertex is different: see the right hand side of Figure~\ref{fig:BlueRelations}.
\end{remark}

\begin{figure}
\input figs/BlueR2.pstex_t \hspace{1.5cm} \raisebox{4mm}{\input figs/BlueR4.pstex_t}
\caption{On the left, the derivation of the blue (opposite surface orientation) R2 relation, which coincides with the black (blackboard oriented) R2 relation. On the right the blue R4 relation, which differs from the black R4 relation by a twist. Note that in the R4 relation, the under strand may be black or blue, and can be directed either way.}\label{fig:BlueRelations}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsubsection{The auxiliary operations of $\wTF$} \label{subsubsec:wops}
The circuit algebra $\wTF$ is equipped with several auxiliary operations.

The first of these is the familiar orientation switch: given a strand $e$ of a w-foam, $\glos{S_e}$ switched the direction (arrow) of the strand $e$.

In the local topological interpretation $S_e$ 
switches both the strand direction and the surface orientation of the strand $e$, as reversing the strand direction reverses the default (black) orientation assigned via the tubing map. 

\parpic[r]{\input{figs/StringUnzip.pstex_t}}
The most interesting operation on w-foams is the {\em strand unzip} $\glos{u_e}$, which doubles the strand $e$ using the
blackboard framing, then attaches the ends of the doubled strand to the
connecting ones, as shown on the right. Unzip is only defined when the orientations of the strands involved match, as shown on the left.  We restrict\footnote{This is a somewhat
artificial condition which we impose to get equations equivalent to
the~\cite{AlekseevTorossian:KashiwaraVergne} equations.} unzip to strands
whose two ending vertices are of different signs. 



\begin{comment}\label{com:wTFFraming}
In order to understand the local topological meaning of the unzip operation, we need to discuss framings in more depth. Recall that
framings were mentioned in Section \ref{subsec:TangleTopology}, but have not played
a significant role so far, except to explain the lack of a Reidemeister 1
relation. 

In the local topological interpretation of $\wTF$, 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. 
Topologically, unzip 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 circle obtained by pushing off any given girth
of the tube in the framing direction is not linked with the original tube,
which is indeed the case.

\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 directions (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}

When a tube is unzipped, 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}. 

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

\begin{center}
\input{figs/CapUnzip.pstex_t}
\end{center}

Note that strands which contain wens may be unzipped by first removing or relocating the wens to other strands using the CW and TV relations.

The strand deletion (denoted $d_e$) operation is restricted to ``long linear'' strands, meaning strands that do not end in a vertex on either side.

Finally, there is a global surface orientation reversal operation $\mu$: this reverses the colour of an entire w-tangled foam, but otherwise does not change the diagram.

%\begin{example} As an example, we derive a relationship between the positive and negative black generating vertices, using circuit algebra compositions and the global mirror operation: see Figure~\ref{fig:Adjoint}.\end{example}

%\begin{figure}
%\input{figs/AdjointV.pstex_t}
%\caption{Deriving the (black) negative vertex from the (black) positive vertex: conjugate the vertex by three wens, apply the TV relation, compose with a virtual crossing, and apply the global mirror operation. Note that the compositions are basic circuit algebra operations.}\label{fig:Adjoint}
%\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\draftcut
\subsection{The Associated Graded Structure} \label{subsec:fgrad}
Mirroring the previous section, we describe the associated graded structure
$\glos{\calA^{sw}}$ of $\wTF$ 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 $\calA^{sw}$''.
\[ \calA^{(s)w}=\CA\!\left.\left.\left\langle
  \raisebox{-6mm}{\input{figs/wTFprojgens@.pstex_t}}
  \right|
  \parbox{1.4in}{\centering $\aft$, TC, VI, CP, $W^2$, TW, CW, FR, (RI for $\calA^{sw}$)}
  \right|
  \parbox{1.0in}{\centering $S_e, u_e, d_e, \mu$}
  \right\rangle.
\]

In other words, $\calA^{(s)w}$ is the circuit algebras of arrow diagrams on trivalent (or foam) skeleta with 
caps and wens. That is, skeleta are elements of $\calS$ as in Section~\ref{subsec:wTF}.  
All but the first three of the generators are skeleton features (of degree 0), and that the single arrows are of degree $1$. As for the generating vertices, the same remark applies as in Definition \ref{def:wTFo},
that is, vertices come all possible combinations of strand directions.

\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 $\calA^{sw}=\calA^w/RI$, arrow
diagrams in $\calA^{(s)w}$ satisfy the following additional relations:

{\it Vertex invariance}, denoted by \glost{VI}, are relations which arise
from the same principle as the classical $\aft$ relation, but with a vertex in place of 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 -- black or blue, positive or negative, with strands oriented in any direction -- 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 vanish next to a cap, as shown on the right. This is true regardless of the colour (surface orientation) of the capped strand. We denote this relation also by
\glost{CP}. (Note that an arrow tail near a cap does not vanish.)

The $W^2$, TV, and CW relations are the skeleton relations introduced in Section~\ref{subsubsec:wrels} which describe the interactions of wens with each other, vertices and caps.


In addition, the Flip Relations FR imply that wens ``commute'' with
arrow heads, but ``anti-commute'' with arrow tails. We also call these \glost{FR}
relations. In the figure below the strand colours are not shown, as it doesn't matter whwther the wens transition from blue to black or black to blue.
\begin{center}
  \input figs/WenRel.pstex_t
\end{center}


\medskip
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, modulo the relations of $\calA^{(s)w}$ and the $\aSTU$ relations of Figure~\ref{fig:STU}, by $\calA^{(s)wt}$. We have the following bracket-rise theorem:

\begin{theorem}\label{thm:FoamBracketRise} The natural 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}

\medskip
As in Section~\ref{subsec:vw-tangles}, the primitive elements of
$\calA^{(s)w}$ are connected diagrams (that is, connected with the skeleton removed), which are linearly generated by trees and wheels. Before
moving on to the auxiliary operations of $\calA^{(s)w}$, we 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}
$\calA^{(s)w}(\FlippedYGraph)=\calA^{(s)w}(\uparrow_2)$, where $\calA^{(s)w}(\FlippedYGraph)$
stands for the space of arrow diagrams whose skeleton is a vertex of any type, with
any orientation of the strands, and $\calA^{(s)w}(\uparrow_2)$
denotes the space of arrow diagrams on two strands.
\end{lemma}

\begin{proof}
 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 $S_e$ changes
the sign of a crossing involving the strand $e$. Hence, letting $\sigma\in \calS$ denote any foam (trivalent) skeleton, the induced (associated graded) arrow diagram
operation is a map $S_e\colon \calA^{(s)w}(\sigma) \to \calA^{(s)w}(S_e(\sigma))$ which acts by reversing the direction of the skeleton strand $e$, and multiplying  
each arrow diagram by $(-1)$ raised to the number of arrow endings on $e$ (counting both heads and tails).

\parpic[r]{\input{figs/Unzip.pstex_t}}
The arrow diagram operations induced by unzip and disc unzip $u_e\colon  \calA^{(s)w}(S) \to \calA^{(s)w}(u_e(S))$ are both denoted $u_e$, and interpreted appropriately according to whether the 
strand $e$ is capped. They both map 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 a primitive arrow diagram $D$ there are $k$ arrow
ends on $e$, then $u_e(D)$ is a sum of $2^k$ primitive arrow diagrams.

The operation induced by deleting the long linear strand $e$ is the map $d_e\colon  \calA^{(s)w}(S) \to \calA^{(s)w}(d_e(S))$ which kills arrow diagrams with
any arrow ending (head or tail) on $e$, and leaves all else unchanged, except with $e$ removed. 


Since $\mu$ does not change the crossing signs, the associated graded operation (also denoted $\mu$) simply reverses the colour of the skeleton without affecting the arrows of the arrow diagram (or the arrow graph of a Jacobi diagram). 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\draftcut
\subsection{The homomorphic expansion}\label{subsec:wTFExpansion}
If a homomorphic expansion for $\wTF$ exists, it is determined by the values of the generators, as $\wTF$ has a finite presetantion. We are interested in particular in group-like homomorphic expansions, where the values of the generators are exponentials of (infinite series of) primitive arrow diagrams. For more detail see \cite[Section2.5.1.2]{Bar-NatanDancso:WKO1}. 

We will see that the value of the (black) positive vertex $V\in \calA^{sw}(\FlippedYGraph)\cong\calA^{sw}(\uparrow_2)$ plays a particularly important role. 
It will also  become clear that the short arrows of $V$ can be ignored: here a short arrow means an arrow on a single strand of the vertex. The reason is that, given a homomorphic expansion $Z$ with $Z(\FlippedYGraph)=V=e^v$, and $a$ is a short arrow on the vertex, then changing $V$ to $V'=e^{v+a}$ defines another homomorphic expansion $Z'$. We explain this in more detail later; for now we make the following definition:

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


Given a homomorphic expansion $Z:\wTF \to \calA^{sw}$, denote by $W$ the $Z$-value of the (blue-to-black) wen. We are now able to state one of the main theorems of this paper:

\begin{theorem}\label{thm:WenATEquivalence}
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 expansions $Z: \wTF \to \calA^{sw}$ exist, and thise which are v-small and satisfy $W=1$ are in one-to one correspondence 
with solutions to the Kashiwara-Vergne equations (defined in Section~\ref{subsec:EqWithAT}) with even Duflo function. \qed
\end{theorem}

Our goal is to explain and prove this theorem. To begin, observe that
finding a homomorphic expansion $Z: \wTF \to \calA^{sw}$ is equivalent to finding values for the generators
of $\wTF$ in $\calA^{sw}$, so that these values satisfy the equations
which arise from the relations in $\wTF$ and the homomorphicity
with respect to the auxiliary operations. 
In this subsection we derive these equations; in Section~\ref{subsec:EqWithAT} we show
that they are equivalent to the Alekseev-Torossian version of the
Kashiwara-Vergne equations \cite{AlekseevTorossian:KashiwaraVergne} with even Duflo function. 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 independently prove this
result in the context of homomorphic expansions for w-tangled foams.

First we set notation for the images of the most important generators. Let $\glos{R}:=Z(\overcrossing) \in \calA^{sw}(\uparrow_2)$. 
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.  Let $\glos{W}\in \calA^{sw}(\uparrow)$ denote the $Z$-value of the blue-to-black wen, and $W'$ the value of the black-to-blue wen, as in Figure~\ref{fig:ZofWen1}. 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
$\glos{V^-}:=Z(\raisebox{-1mm}{\input{figs/MinusVertex.pstex_t}})\in
\calA^{sw}(\raisebox{-1mm}{\input{figs/MinusVertex.pstex_t}})\cong
\calA^{sw}(\uparrow_2)$. 

\begin{lemma}\label{lem:ZandMu}
For any generator $g$ and homomorphic expansion $Z:\wTF \to \calA^{sw}$, the value $Z(g)$ and the value of the opposite colour generator $Z(\mu(g))$ only differ in the colour of the skeleton.
\end{lemma}

\begin{proof}
This is an immediate consequence of the homomorphicity of $Z$ with respect to $\mu$: for any $F\in \wTF$, we have $Z(\mu(F))=\mu(Z(F))$, and $\mu$ on $\calA^{sw}$ changes the skeleton colouring only, not the arrows.\qed
\end{proof}


We first address the value of the wen. Assume that $Z$ is a homomorphic expansion fro $\wTF$, and let $W$ denote the value of the ``blue-to-black'' wen, and $W'$ the value of the ``black-to-blue'' wen, as shown in Figure~\ref{fig:ZofWen1}. 

Recall that the FR relation in $\calA^{sw}$ states that skeleton wens commute with arrow heads and anti-commute with arrow tails. For a primitive arrow diagram $D$ we denote
$$\overline{D}:= (-1)^{\#\{ \text{arrow tails in }D\}} D.$$
Then we have that $wD=\overline{D}w$, where $w$ denotes a skeleton wen and $D\in \calA^{sw}(\uparrow)$. Though note the abuse of notation: technically in this equality $D$ and $\overline{D}$ are on differently coloured skeleta. 
The same equality holds in $\calA^{sw}(\uparrow_n)$ if all strands of $D$ are commuted with wens.



\begin{figure}
\input figs/ZofWen1.pstex_t
\caption{The values of the wens under a homomorphic expansion $Z$.}\label{fig:ZofWen1}
\end{figure}



\begin{lemma}\label{lem:ValueW}
Under any group-like homomorphic expansion $Z$ the values of the wen can be expressed as $W=W'=\exp({\sum_{k=1}^\infty c_{2k+1}w_{2k+1}})$, where $c_{2k+1}$ are constants, and $w_{2k+1}$ are odd wheels, and any such $W$ satisies the equations induced by the $W^2$ relation and homomorphicity with respect to $\mu$ and $S_e$.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:ZandMu}, we have $W=W'$, that is, $W$ and $W'$ agree as arrow graphs, although they live on differently coloured skeleta. Thus, 
from the $W^2$ relation we obtain that $\overline{W}W=1$, see Figure~\ref{fig:ZofWen2}. 

Since $Z$ is group-like, we have $W=e^\omega$ for some primitive $\omega$. Since $\omega$ is a primitive element of $\calA^{sw}(\uparrow)$, by the description of primitive arrow diagrams in Section~\ref{subsec:ATSpaces} it can be written a sum of wheels in degrees 2 and above, with possibly a multiple of a single arrow in degree 1. (Higher degree trees on a single strand reduce to wheels by the AS and STU relations.) Write $\omega= p_1a + \sum_{k=1}^{\infty} p_{2k}w_{2k} + \sum_{l=1}^\infty p_{2l+1}w_{2l+1}$, where $a$ denotes the degree 1 arrow, $w_i$ are $i$-wheels, and $p_i$ are constants.  

Then $\overline{W}=e^{\overline{\omega}}$, and $\overline{\omega}= - p_1a + \sum_{k=1}^{\infty} p_{2k}w_{2k} - \sum_{l=1}^\infty p_{2l+1}w_{2l+1}$. Thus, $\overline{W}=W^{-1}$ means that $p_{2k}=0$ for all $k\geq 1$, in other words, $W$ is contained in odd degrees only.


From the homomorphicity of $Z$ with respect to orientation switches we further see that $S(W)=\overline{W'}$, where $S(W)$ denotes the orientation switch of $W$. Combining this with the previous result, we have $S(W)=\overline{W}$, which further implies that $p_1=0$, completing the proof. \qed
\end{proof}


\begin{figure}
\input{figs/ZofWen2.pstex_t}
\caption{The implication of the $W^2$ relation.}\label{fig:ZofWen2}
\end{figure}



\medskip
Recall that by convention we number strands at the bottom of each diagram from left to right, and for an arrow diagram $D\in \calA^{sw}(\uparrow_k)$, $D^{i_1i_2\ldots i_k}$ means ``$D$ placed on strands $i_1,\ldots,i_k$. For instance, $R^{23}$ means ``$R$ placed on strands 2 and 3''. In this section 
we also need to use co-simplicial notation, for example $R^{(23)1}$ means ``$R$ with its first strand doubled (unzipped), then placed on strands 2, 3 and 1''.


\begin{lemma}\label{lem:ValueR}
For any homomorphic expansion $Z$, the values of the crossings are as follows: $R=Z(\overcrossing)=e^a$ where $a$ denotes a single arrow from the over to the under strand, $Z(\undercrossing)=(R^{-1})^{21}=Z^{-a^{21}}$, where again $a^{21}$ points from the over strand to the under strand. The same formulas hold for positive (resp.~negative) crossings of any colour. These values satisfy the equations induced by $R1^s$, $R2$, $R3$ and $OC$ in $\calA^{sw}$.
\end{lemma}

\begin{proof}
This is a small extension of the results already established in Sections
\ref{subsec:vw-tangles} and \ref{subsec:UniquenessForTangles}. We showed there that
R3 with group-likeness and homomorphicity implies that $R=e^a$, and  
it follows from the Reidemeister 2 relation that
$Z(\undercrossing)=(R^{-1})^{21}$. 

The values of the blue and mixed-colour crossings can be derived from $R$ using the homomorphicity of $Z$ with respect to the global symmetry $\mu$, and the flip relations. An example of these calculations is shown in Figure~\ref{fig:FRCalc}. 

These values also automatically satisfy the equation arising from the $R1^s$ relation, and satisfy the equation induced by the $OC$ relation due to the $TC$ relation in $\calA^{sw}$ \qed
\end{proof}
\medskip 



\begin{figure}
{
  \def\ol{\overline}
  \def\Rea{$R=e^a$}
  \def\RbarR{$\ol{R}\!=\!e^{-a}\!=\!R^{-1}$}
  \def\BoB{{\raisebox{-1.75mm}{\input{figs/BlueOverBlackMinus.pstex_t}}}}
  \def\ZBoB{$Z\left(\BoB\right)^{21}$}
  \begin{center} \input{figs/FRCalc.pstex_t} \end{center}
   \begin{center} $(R^{-1})^{21} = Z\left(\BoB\right)$ \end{center}
}
\caption{Deriving the value of the negative vertex with blue over-strand, using the FR relation.}\label{fig:FRCalc}
\end{figure}


\begin{lemma}\label{lem:R4}
The R4 relations induce the following single equation on $V$ and $R$:
\begin{equation}\tag{R4}\label{eq:HardR4}
 V^{12}R^{(12)3}=R^{23}R^{13}V^{12}.
\end{equation}
\end{lemma}

\begin{proof}
The Reidemeister 4
relation with a strand over a vertex induces an equation that is automatically satisfied, as follows:
\begin{center}
 \input{figs/R4ToEquation.pstex_t} 
\end{center}
In other words, the over strand $R4$ relation induces the equation
$$V^{12}R^{3(12)}=R^{32}R^{31}V^{12}.$$
However, observe that by the ``head-invariance'' property of arrow diagrams (Remark \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 an application of the TC relation ($a^{31}$ and $a^{32}$ commute). Therefore, this equation is true regardless of the choice of $V$.

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

\medskip


\begin{lemma}\label{lem:CP}
The equation induced by the CP relation is automatically satisfied for any choice of $C$.
\end{lemma}
\begin{proof}
The Cap Pull-out (CP) relation translates to the equation $R^{12}C^2=C^2$. By head invariance,
$R^{12}C^2=C^2R^{12}$. Now $R^{12}$ is just below the cap on strand $2$, and thus by the CP relation in $\calA^{sw}$, every term of $R^{12}$
with an arrow head at the top of strand $2$ is zero. Hence, the only surviving term of $R^{12}$ is $1$ (the empty diagram), which makes the
equation true.
\end{proof}


\begin{lemma}\label{lem:VV-}
Assume that $Z: \wTF \to \calA^{sw}$ is a group-like homomorphic expansion, and denote by $V:=Z(\FlippedYGraph)$ the value of the black positive vertex, and $V^-:=Z(\YGraph)$ the value of the black negative vertex. Then $V^-=V^{-1}$.
\end{lemma}

\begin{proof}
This is an immediate consequence of the homomorphicity of $Z$ with respect to the unzip operation. \qed
\end{proof}
\medskip

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

\begin{lemma}\label{lem:CW}
Let $C=e^c$, where  The equation induced by the CW relation is $c_{odd}=-\frac{1}{2}\omega$, or equivalently $C_{odd}=W^{-1/2}$.
\end{lemma}

\begin{proof}
Applying $Z$ to each side of the CW relation, we obtain $\overline{W}\overline{C}=C$ in $\calA^{sw}(\upcap)$. Substituting the formulas for $W$ and $C$, the statement follows.\qed
\end{proof}

\medskip


Possibly the most interesting equation is the one induced by the twisted vertex relation. For this we introduce one additional piece of notation. Given $D\in \calA^{sw}(\uparrow_n)$, denote $\glos{A(D)}:=S_1S_2\cdots S_n(\overline{D})$, and call this the {\em adjoint} of $D$. In other words, the operation 
$A: \calA^{sw}(\uparrow_n) to \calA^{sw}(\uparrow_n)$ 
reverses the strand directions and multiplies an arrow diagram $D$ by 
$(-1)^{\#{\{\text{arrow heads on the skeleton}}\}}$ .


\begin{lemma}\label{lem:WenUnitarity}
The TV relation is induces the ``Wen-Unitarity'' equation   
\begin{equation}\tag{WU}\label{eq:WU}
(W^{-1})^{(12)} A(V)W^1W^2V=1.
\end{equation}
\end{lemma}

\begin{proof}
We apply $Z$ to each side of the TV relation, as shown in Figure~\ref{fig:TVUnitarity}.
On the right hand side of the relation is a negative vertex $\raisebox{-1mm}{\input{figs/MinusVertex.pstex_t}}$ with the strand orientations reversed, upside down and the strands numbered $(2,1)$ as the vertex follows a virtual crossing. Therefore, the value of this vertex is $S_1S_2(V^{-1})$ by Lemma~\ref{lem:VV-}. 

The top wen on the far left side can be ``pulled down'' to the bottom two strands using the VI relation. Therefore, we obtain the following equation in $\calA^{sw}(\uparrow_2)$:
$$\overline{W}^1\overline{W}^2 \overline{V}W^{(12)}=S_1S_2(V^{-1})$$

Applying $S_1S_2$ to both sides, and using from Lemma~\ref{lem:ValueW} that $\bar{W}=W^{-1}=S(W)$, we obtain the equation \eqref{eq:WU}. \qed
\end{proof}
\medskip

\begin{figure}
{
  \def\SSV{$S_1S_2(V^{-1})$}
  \def\OV{$\overline{V}$}
  \def\OW{$\overline{W}$}
  \begin{center} \input{figs/TVUnitarity.pstex_t} \end{center}
}
\caption{Applying $Z$ to each side of the TV relation.}\label{fig:TVUnitarity}
\end{figure}


\begin{lemma}\label{lem:CapEq} 
Homomorphicity of $Z$ with respect to the disc unzip operation is equavialent to the Cap Equation:
\begin{equation}\tag{C}\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}
\end{lemma}


\parpic[r]{\input{figs/CapEqn.pstex_t}}\picskip{5}
\noindent {\em Proof.} We need to apply $Z$ and the cap unzip $u$ in either order to the w-foam shown in the figure on the right. On the left hand side, the value of the cap is unzipped and gives $C^{(12)}$. Note that \eqref{eq:CapEqn} is an equation in $\calA^{sw}(\upcap_2)$. \qed

\vspace{1cm}

To summarize, we have proven the following theorem:

\begin{theorem}\label{thm:ZWenEquations}
 $Z:\wTF \to \calA^{sw}$ is a group-like homomorphic expansion if and only if the values of the key generators $R=e^a$, $V$, $W$ and $C$ are group-like, and satisfy the equations
\begin{enumerate}
\item[(R4)]  $V^{12}R^{(12)3}=R^{23}R^{13}V^{12}$ in $\calA^{sw}(\uparrow_3)$,
\item[(WU)] $(W^{-1})^{(12)} A(V)W^1W^2V=1$ in $\calA^{sw}(\uparrow_2)$,
\item[(C)] $V^{12}C^{(12)}=C^1C^2$ in
  $\calA^{sw}(\upcap_2)$,
\item[(CW)] $C_{odd}=W^{-1/2}$ in $\calA(\upcap_1)$, or equivalently, as power series in odd wheels.
\end{enumerate}
\end{theorem}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\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, called the {\em Duflo function}. 
In other words, the second condition says that there exists 
$a \in \attr_1$ such that $jF=a(x)+a(y)-a(\log(e^{x}e^{y}))$.

\medskip

\noindent{\em Proof of Theorem~\ref{thm:WenATEquivalence}.}
We need to translate the equations of Theorem~\ref{thm:ZWenEquations} to equations in the Alekseev-Torossian spaces, using the identifications of Proposition~\ref{prop:Pnses} and the identification of wheels with cyclic words. Note the condition in Theorem~\ref{thm:WenATEquivalence} that $W=1$. The (CW) equation now simply asserts that the value $C$ is an even power series in wheels. The \eqref{eq:WU} equation simplifies to the following, which we call the {\em Unitarity} of the vertex:
\begin{equation}\tag{U}\label{eq:unitarity}
A(V)V=1.
\end{equation}


Recall from Section \ref{subsec:ATSpaces} that the map $u\colon  \tder_2 \to \calA^{sw}(\uparrow_2)$ plants the head of a tree
above all of its tails.
Suppose that the values $V$ and $C$ satisfy the simplified equations of Theorem~\ref{thm:ZWenEquations} with $W=1$. Write  $V=e^be^{uD}$, where $b \in \tr_2^s$, $D \in  \tder_2\oplus \fraka_2$, 
and where  $V$ can be written in this form without loss of generality because wheels can always be commuted to the bottom of a diagram (at the possible cost of more wheels). Furthermore, $V$ is group-like and hence it can be written in exponential form. Similarly, write $C=e^c$ with $c \in \attr_1^s$.

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, $Z$ is v-small and $D \in \tder_2$. 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 $c$ is already assumed to be even.)  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, it remains unchanged if we multiply the left side on the right by $e^{-uD}$: $uD$ has its head at the top, so it
is 0 by the Cap relation, hence $e^{uD}=1$ near the cap. Hence, $$e^be^{uD}e^{c(x+y)}e^{-uD}=e^{c(x)+c(y)}.$$ 

\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 linear 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 even Duflo function $a \in \tr_1$ 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 the even cap value $C=e^c$ satisfy the equations for 
homomorphic expansions (\ref{eq:HardR4}), (\ref{eq:unitarity})
and (\ref{eq:CapEqn}), and hence define a homomorphic expansion of $\wTF$ with $W=1$.

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


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



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Orientable w-tangled foams}\label{subsec:OriFoams}

There is a one-coloured sub-circuit algebra of $\wTF$ consisting of the w-tangled foams which are {\em all solid black}: in other words, contain no wens, and equipped with the positive surface orientation. We call this the circuit algebra of orientable w-foams, and denote it by $\wTFo$. 

\begin{lemma}\label{lem:CancelWens}
If $F\in \wTF$ is a w-foam all of whose tangle ends are the same colour, then all of the wens in $F$ cancel by the wen relations. Furthermore, the process of cancelling all wens can be made canonical by a choice -- for each connected component of the skeleton of $F$ -- of 
a spanning tree $T$, and  a basepoint on $T$, which is a tangle end if there are any. 
\end{lemma}

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

The obvious but key observation is that there are an even number of wens along any cycle in $\sigma(F)$, as well as any path connecting two tangle ends. Given a choice of spanning tree $T$ for $\sigma(F)$ and a base point on it, there is a unique way to ``clear $T$ of wens''. Namely, use the TV relation to push wens off of $T$ away from the base point. At the end of this process, all wens will end up either at a tangle end (which are all necessarily in $T$, or on an edge of of $\sigma(F)$ not in $T$.

Since there are an even number of wens on the path from any given tangle end to the base point (which is also necessarily a tangle end in this case), there is an even number of wens at each tangle end at the end of the process, therefore they cancel by the $W^2$ relation. For any non-$T$ edge $e$ of $\sigma(F)$, there is a unique path $\gamma$ in $T$ which connects the two ends of $e$. Since there originally was an even number of wens along the cycle $e \cup \gamma$, there is an even number of wens on $e$ at the end of the process, which therefore cancel. \qed
\end{proof}
\medskip

We derive a generators - relations - operations presenation for $\wTFo$. Since the wen is no longer a generator, there are no wen relations. The operations $S_e$, $u_e$ and $d_e$ restrict to $\wTFo$. The operation $\mu$ does not, however there is a composite operation which does make sense, as follows. For a w-foam $F\in \wTFo$, consider $F$ as an element of $\wTF$, and let $F^w$ denote $F$ composed with a wen at every tangle end (call this ``wenjugation''). By Lemma~\ref{lem:CancelWens} all these wens cancel, thus, $F^w$ is equivalent to an entirely hollow-blue w-foam with no wens. Therefore, $\mu(F^w)\in \wTFo$ again. We denote $\overline{F}:= \mu(F^w)$, and replace $\mu$ with this operation $-:\wTFo \to \wTFo$.  

\begin{definition}\label{def:wTFo} The circuit algebra of oriented w-foams is given by the presentation
\[
  \wTFo=\CA\!\left.\left.\left\langle
  \raisebox{-2mm}{\input{figs/wTFgens.pstex_t}}
  \right|
  \parbox{1.9in}{$R1^s$, R2, R3, R4, OC, CP} 
  \right|
  \parbox{0.8in}{$S_e,u_e,d_e,-$}
  \right\rangle.
\]
\end{definition}

While not strictly necessary for our main result, we verify for interest that $\wTFo$ as in the definition above injects into $\wTF$. In other words, the generators and relations description above is indeed a description of the black uni-coloured sub-circuit algebra of $\wTF$. 

\begin{proposition}
The circuit algebra of oriented w-foams $\wTFo$ injects into $\wTF$.
\end{proposition}

\begin{proof}
We need to show that given $F, F' \in \wTFo$ for which $F~F'$ via a sequence of $\wTF$ relations, then $F~F'$ also in $\wTFo$. This can be verified explicitly, as follows. Choose a spanning tree and base point for each connected component of $\sigma(F)=\sigma(F')$. Let $F=F_0\sim F_1\sim \cdots \sim F_n=F'$ be a sequence of $\wTF$ moves. Via the process of Lemma~\ref{lem:CancelWens}, each $F_i$ (i=0,\ldots, n) is canonically equivalent to an element of $\wTFo$, call this element $\Omega(F_i)$. Hence, we only need to show that $\Omega(F_i)\sim \Omega(F_{i+1})$ in $\wTFo$, where $F_i$ and $F_{i+1}$ differ in a single relation in $\wTF$. This is obvious if that relation is not a wen relation, easy for the $W^2$ and $CW$ relations, and can be directly verified with some effort for the $FR$ and $TV$ relations.\qed
\end{proof}
\medskip

The circuit algebra $\wTFo$ is again skeleton graded, with skeleton circuit algebra given by
\[
  \calS^0 = \CA\!\left\langle\raisebox{-2mm}{\input{figs/SkelGen.pstex_t}}\right\rangle
\]

The associated graded structure -- which we continue to denote by $\calA^{sw}$ to avoid too many superscripts -- consists of arrow diagrams on uni-coloured skeleta (elements of $\calS^o$), given by the presentation
 \[ \calA^{sw}=\CA\!\left.\left.\left\langle
  \raisebox{-2mm}{\input{figs/wTFprojgens.pstex_t}}
  \right|
  \parbox{1.5in}{\centering $\aft$, TC, VI, CP, RI}
  \right|
  \parbox{1in}{\centering $S_e, u_e, d_e, -$}
  \right\rangle.
\]
Here $-$ is the associated graded operation of $-:\wTFo \to \wTFo$, and it coincides with the operation $D\mapsto \overline{D}$ defined in Section~\ref{subsec:wTFExpansion}, which multiplies the arrow diagram $D$ with $(-1)^{\#\{\text{arrow tails}\}}$.
Arrow diagrams again have an alternative, equivalent description in terms of Jacobi diagrams, as in Theorem~\ref{thm:FoamBracketRise}.

Our main theorem states that homomorphic expansions for $\wTFo$ are in bijection with Kashiwara-Vergne solutions, without restriction on the Duflo function:

\begin{theorem}
\label{thm:ATEquivalence} There exist a group-likehomomorphic expansions for $\wTFo$, and
there is a bijection
between the set of solutions $(F,a)$ of the generalized KV equations \eqref{eq:ATKVEqns} and the set of v-small group-like homomorphic expansions
for $\wTFo$.
\end{theorem}

\begin{proof}
Since there are no wens, a homomorphic expansion is determined by the values $R$, $C$, and $V$, with  $Z(\undercrossing)=(R^{-1})^{21}$ by the R2 relation, and $Z(\raisebox{-1mm}{\input{figs/MinusVertex.pstex_t}})=V^{-1}$ by the homomorphicity with respect to edge unzip.

We derive $R=e^a$, and the \eqref{eq:HardR4} and \eqref{eq:CapEqn} equations as before: from the R3, R4 and CP relations, and the homomorphicity with respect to $S_e$ and $d_e$.


\qed
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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.
