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\title{{Finite Type Invariants
  of w-Knotted Objects III: w-foams, the Kashiwara-Vergne Theorem and Drinfel'd associators}}

\author{Dror~Bar-Natan}
\address{
  Department of Mathematics\\
  University of Toronto\\
  Toronto Ontario M5S 2E4\\
  Canada
}
\email{drorbn@math.toronto.edu}
\urladdr{http://www.math.toronto.edu/~drorbn}

\author{Zsuzsanna Dancso}
\address{
  Mathematical Sciences Institute\\
  Australian National University\\
  John Dedman Bldg 26\\
  Acton ACT 2601, Australia
}
\email{zsuzsanna.dancso@anu.edu.au}
\urladdr{http://www.math.toronto.edu/zsuzsi}

\date{first edition in future, this edition \today. The
\arXiv{????.????} edition may be older}

\subjclass{57M25}
\keywords{
  virtual knots,
  w-braids,
  w-knots,
  w-tangles,
  knotted graphs,
  finite type invariants,
  Kashiwara-Vergne,
  associators,
  double tree,
  free Lie algebras%
}

\thanks{The first author's work was partially supported by NSERC grants RGPIN-264374 and RGPIN-2018-04350, and by the Chu Family Foundation (NYC), and wishes to thank the Sydney Mathematics Research Institute for their hospitality and support.
The second author was partially supported by NSF grant no.~0932078~000 while in residence
at the Mathematical Sciences Research Institute, and by the Australian Research Council DECRA DE170101128 Fellowship.
  Electronic version and related files at~\cite{DoubleTree},
  \url{http://www.math.toronto.edu/~drorbn/papers/WKO3/}.
}

\begin{abstract}
  \input abstract.tex
\end{abstract}

\maketitle
 
\tableofcontents

\section{Introduction}

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

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

\begin{itemize}
 
\item For many classes of knotted objects in 3-dimensional spaces homomorphic expansions don't exist --- for example, one would have loved ordinary tangles to have homomorphic expansions, but they don't. 

\item Yet  {\em parenthesized tangles}, or nearly-equivalently, {\em knotted trivalent graphs} in $\bbR^3$ -- which, for the purpose of this short summary, we denote by $\calK^u$, denoted $\sKTG$ later -- do have homomorphic expansions. A homomorphic expansion $Z^u\colon\calK^u\to\calA^u$ is defined by its values on a couple of elements of $\calK^u$ which generate $\calK^u$ using the operations $\calK^u$ is equipped with. The most interesting of these generators is the tetrahedron $\tetrahedron$, and $\Phi=Z^u(\tetrahedron)$ turns out to be equivalent to a {\em Drinfel'd associator}.

\item A certain class of four-dimensional knotted graphs called {\em w-foams} -- denoted  $\calK^w$ for now, $\wTF$ later in the paper  -- also has homomorphic expansions. The most interesting generator of $\calK^w$ is the {\em vertex} $\FlippedYGraph$, and if $Z^w\colon\calK^w\to\calA^w$ is a homomorphic expansion, then it turns out that $V=Z^w(\FlippedYGraph)$ is equivalent to a solution of the {\em Kashiwara-Vergne problem} in Lie theory. 
 
\end{itemize}

\parpic[r]{\input{figs/uw.pstex_t}}
Roughly speaking, $\calK^u$ is a part of $\calK^w$ and $\calA^u$ is a part of $\calA^w$, as in the figure on the right. More precisely, there are natural maps $a\colon\calK^u\to\calK^w$ and $\alpha\colon\calA^u\to\calA^w$. The main purpose of this paper is to prove the following theorem, whose precise version is stated later as Theorem~\ref{thm:main}: 

\par\noindent{\bf Theorem.} Any homomorphic expansion $Z^u$ for $\calK^u$ extends uniquely to a homomorphic expansion $Z^w$ for $\calK^w$, and therefore, any Drinfel'd associator $\Phi$ gives rise to a solution $V$ of the Kashiwara-Vergne problem.

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

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

\[ \input{figs/QuickDT.pstex_t} \]



%----------------------------------------------------------------------------------------------------




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

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

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

In \cite{AlekseevTorossian:KashiwaraVergne} Alekseev and Torossian gave another proof of the KV conjecture based on a deep connection with Drinfel'd associators. In turn, Drinfel'd's theory
of associators~\cite{Drinfeld:QuasiHopf} can be interpreted as a theory of well-behaved expansions (universal finite type
invariants) of parenthesized tangles\footnote{``$q$-tangles'' in~\cite{LeMurakami:Universal}, ``non-associative tangles''
in~\cite{Bar-Natan:NAT}.}~\cite{LeMurakami:Universal, Bar-Natan:NAT}, or of knotted trivalent graphs~\cite{Dancso:KIforKTG}.
In \cite{AlekseevEnriquezTorossian:ExplicitSolutions} Alekseev, Enriquez and Torossian gave an explicit formula
for solutions of the Kashiwara-Vergne equations in terms of Drinfel'd associators.

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

The algebraic structure of w-foams is built on the concept of a {\em circuit algebra}, which was also identified as equivalent to the operadic structure of a {\em wheeled prop} in \cite{DHR:CircAlg}. The symmetry groups of Kashiwara-Vergne solutions, called the Kashiwara-Vergne groups, are shown to be automorphism groups of the w-foam circuit algebra and its associated graded arrow diagrams in \cite{DHR:KVKRV}. The relationship between the symmetries of Drinfel'd associators -- the Grothendieck-Teichmuller groups -- and the Kashiwara-Vergne groups is described in the topological context of w-foams in \cite{DHaR:GRTKRV}.

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


\subsubsection{Topology}
We begin by describing a chain of maps from ``parenthesized braids'' to ``(signed) knotted trivalent graphs'' to ``w-tangled foams'':
$$\calK:=\{\uPB \stackrel{\cl}{\longrightarrow} \sKTG \stackrel{a}{\longrightarrow} \wTFe\}.$$

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

\begin{figure}
 \input{figs/PBexample.pstex_t}
 \caption{Two examples of parenthesized braids. Note that by convention the parenthetization can be read from the distance scales between the endpoints
 of the braid, and so we omit the parentheses in parts of this paper.}\label{fig:PBexample}
\end{figure}

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

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

The map $\cl: \uPB \to \sKTG$ is the ``closure map''. Given a parenthesized braid, close up its top and bottom each by 
gluing a binary tree according to the parentetization;
this produces a $\sKTG$ with the convention that all strands are oriented upwards, bottom vertices are negative, and top vertices are positive.
An example is shown below.
\begin{equation}\label{eq:cl}
\raisebox{-1.5cm}{\input{figs/Closure.pstex_t} }
\end{equation}

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

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

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

Associated graded structures are 
taken with respect to the filtration by powers of the {\em augmentation ideal}. For the spaces $\uPB$, $\sKTG$ and $\wTFe$, the 
associated graded spaces $\calA^{hor}$, $\calA^{u}$ and $\calA^{sw}$ are the spaces of ``horizontal chord diagrams on parenthesized strands'', ``chord diagrams
on trivalent skeleta'', and ``arrow diagrams'', as described in \cite{Bar-Natan:GT1}, \cite[Section 4.6]{Bar-NatanDancso:WKO2}, and Section \ref{sec:wTFe} 
of this paper, respectively. Thus, the associated graded structure of $\calK$ is
$$\calA:=\{\calA^{hor}\stackrel{\cl}{\longrightarrow} \calA^{u} \stackrel{\alpha}{\longrightarrow} \calA^{sw}\},$$
where $\cl$ and $\alpha$ are the maps induced by $\cl$ and $a$, respectively. More specifically, $\cl$ is the ``closure of chord diagrams'', and $\alpha$ is 
``replacing each chord with the sum of its two possible orientations'', see \cite[Section 3.3]{Bar-NatanDancso:WKO2}.

An expansion \cite[Section 2.3]{Bar-NatanDancso:WKO2} is a filtered map from an algebraic structure to its 
associated graded structure, whose associated graded map is the identity. In knot theory, expansions are also known as universal finite type invariants. A homomorphic expansion is an expansion which behaves well with respect to
the operations of the algebraic structure, that is, it intertwines each operation with its induced counterpart on the associated graded structure. (For a more detailed introduction see \cite[Section 2.3]{Bar-NatanDancso:WKO2}.)
Hence, a homomorphic expansion $Z: \calK \to \calA$ is a triple of homomorphic expansions $Z^b, Z^u,$ and $Z^w$ for $\calK^b:=\uPB$, $\calK^u:=\sKTG$ and $\calK^w:=\wTFe$, respectively, so that 
the following diagram commutes: 
\begin{equation}\label{eq:MainDiag}
 \xymatrix{
\calK:\ar[d]^{Z}& \calK^b  \ar[r]^{\cl}  \ar[d]^{Z^b} 
 & \calK^u  \ar[r]^{a}  \ar[d]^{Z^u} 
 & \calK^w  \ar[d]^{Z^w} \\
\calA:& \calA^{hor}  \ar[r]^{\cl} 
 & \calA^u  \ar[r]^{\alpha} 
 & \calA^w
 }
\end{equation} 

\subsubsection{The main result}
We recall \cite{Bar-Natan:GT1} that a homomorphic expansion $Z^b$ for parenthesized braids is determined by a {\em horizontal chord associator} or {\em Drinfel'd associator}
$\Phi=Z^b(\raisebox{-1mm}{\input{figs/Phi.pstex_t}})$.
A homomorphic expansion $Z^u$ of $\sKTG$ is also determined\footnote{With the exception of some minor normalization, see \cite{Bar-NatanDancso:WKO2}, Lemma~4.14 and the paragraph after.} by a Drinfel'd associator (horizontal chords or not; see \cite[Section 4.6]{Bar-NatanDancso:WKO2}),
so the significance of the left commutative square is to force the associator corresponding to $Z^u$ to be a horizontal chord associator. 
In turn, $Z^w$ is determined by a solution $F$  to the Kashiwara-Vergne problem (see \cite[Section 4.4 -- 4.5]{Bar-NatanDancso:WKO2}).
The goal of this paper is to prove the following theorem, which, via the correspondence above, implies the Kashiwara-Vergne Theorem:

\begin{theorem}\label{thm:main} 

\begin{enumerate}
 \item Assuming that $Z: \calK \to \calA$ exists, it is determined\footnote{In fact, almost entirely determined by $Z^b$, with the exception of some minor normalization of $Z^u$ which is not determined by an associator.} by $Z^u$.  
 \item Let $C=Z^w(\downcap)$, then $C$ is given by a power series with even part $C_0=\alpha(\nu^{1/4})$, where $\nu$ is the Kontsevich integral of the unknot. There is a formula for $V=Z^w(\raisebox{-1mm}{\input{figs/PlusVertex.pstex_t}}))$ in terms of $C$ and the Drinfel'd associator $\Phi$ associated to $Z^{u}$: 
\begin{equation}\label{eqn:AET}
V=C^{-1}_{1}C^{-1}_{2}\varphi\left(\Phi^{{-1}}(a_{2(13)},-a_{2(13)}-a_{4(13)})\cdot e^{a_{23}/2}\Phi(a_{23},a_{43})\right)C_{(12)},
\end{equation}
where $a$ denotes a single arrow\footnote{The notation is explained in detail in Section~\ref{subsec:AETFormula}}.
This agrees\footnote{The two formulas are written in different languages, and checking that they agree takes effort. See Section~\ref{subsec:AETFormula} and Appendix~\ref{app:AET}.} with the formula proven in \cite{AlekseevEnriquezTorossian:ExplicitSolutions}.
 \item Every $Z^b$ extends to a $Z$.
\end{enumerate}
\end{theorem} 

\begin{remark}
The formula in part (2) of the Theorem, expresses $V=Z^w(\raisebox{-1mm}{\input{figs/PlusVertex.pstex_t}}))$ in terms of the Drinfel'd associator $\Phi$, and $C=Z^w(\downcap)$. One might wonder if there are separate formulas for both $V$ and $C$ in terms of $\Phi$. In fact, in Corollary \ref{cor:CapValue} we compute the ``even part'' of $C$ explicitly, and show that it is fixed, in other words, does not depend on $\Phi$. Part (1) of the Theorem, proven in Section~\ref{subsec:Part1Proof}, shows that the complete value of $C$ (even and odd part) is uniquely determined by $\Phi$. In Theorem~\ref{thm:Unitarity} we also prove that a suitable value exists for any choice $\Phi$.
\end{remark}

The key to the proof of the theorem is to show that the generator \raisebox{-.7mm}{\input{figs/PlusVertex.pstex_t}} of $\wTFe$ 
can be expressed in terms of the 
generator \raisebox{-.7mm}{\input{figs/Phi.pstex_t}} of $\uPB$ and the operations of $\calK$. Assuming that $Z$ exists, this yields a formula for $V$ in terms
of $\Phi$. 

%To preview the expression of \raisebox{-.7mm}{\input{figs/PlusVertex.pstex_t}} in terms of \raisebox{-.7mm}{\input{figs/Phi.pstex_t}}, let us display a picture with no explanation, below. 
%\begin{center} 
 %\input{figs/PhiToV.pstex_t}
%\end{center}

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

\subsection{Paper Structure} 
In Section~\ref{sec:wTFe} we provide an overview of the space $\wTF$ of (oriented) w-foams and its extension with strings $\wTFe$. We provide a brief review of definitions and crucial facts from \cite{Bar-NatanDancso:WKO2}, and details of the extension. We prove that homomorphic expansions for $\wTF$ extend uniquely to homomorphic expansions for $\wTFe$.

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

Section~\ref{sec:Rmks} is a short section of closing remarks, and in Appendix~\ref{app:AET} we give an explicit comparison and equivalence between our formula in Part (2) and the ALekseev--Enriquez--Torossian \cite{AlekseevEnriquezTorossian:ExplicitSolutions} formula.



%-----------------------------------------------------------------------------------------------------------------



\section{The spaces $\wTFe$ and $\calA^{sw}$ in more detail}\label{sec:wTFe}

As mentioned in the introduction, $\wTFe$ is a minor extension of the space $\wTF$ studied in \cite[Section 4.1 -- 4.4]{Bar-NatanDancso:WKO2}. 
It can be introduced
as a planar algebra or as a circuit algebra; we will do the latter as it is simpler and more concise. Circuit algebras are defined in 
\cite[Section 2.4]{Bar-NatanDancso:WKO2}; in short, they are similar to planar algebras but without the planarity requirement for ``connecting strands''. 
As in \wko, each generator and relation of $\wTFe$ has a local topological interpretation. 
Recall from \cite[Sections 1.2, 3.4, 4.1]{Bar-NatanDancso:WKO2} that $\wTF$ diagrams represent certain ribbon knotted tubes with foam vertices in $\bbR^4$, 
and the circuit algebra $\wTF$ is conjecturally a Reidemeister theory for this space (i.e., there is a surjection $\delta$ from the
circuit algebra $\wTF$ to ribbon knotted tubes with foam vertices, and $\delta$ is conjectured to be an isomorphism). 
The space $\wTFe$ extends 
$\wTF$ by adding one-dimensional strands to the picture. Note that in themselves, one dimensional strands in $\bbR^4$ are never knotted, however, they can be knotted 
{\em with} the two-dimensional tubes. In figures two-dimensional tubes will be denoted by thick lines and one dimensional strings by thin red lines.
With this in mind, we define $\wTFe$ as a circuit algebra defined in terms of generators and relations, and with some extra operations beyond 
circuit algebra compositions. Each generator, relation and operation has a local topological interpretation which provides much of the intuition
behind the proofs. However, the corresponding Reidemeister theorem is only conjectural.


\[
  \wTFe=\CA\!\left\langle
  \raisebox{-8mm}{\input{figs/wTFgens.pstex_t}}
  \left|
  \parbox{0.8in}{\centering relations as in Section~\ref{subsec:wrels}}
  \right|
  \parbox{1.1in}{\centering auxiliary operations as in Section~\ref{subsec:wops}}
  \right\rangle
\]







\subsection{The generators of $\wTFe$}\label{subsec:wgens}
We begin by discussing the local topological meaning of each generator shown above.

\parpic[l]{\includegraphics[width=1.5cm,height=5cm]{figs/TheVertex.ps}}
The first five generators are as described in \cite[Sections 4.1.1]{Bar-NatanDancso:WKO2}, we briefly recall their descriptions here.   
Knotted (more precisely, braided) tubes
in $\bbR^4$ can equivalently be thought of as movies of flying rings in $\bbR^3$. The two crossings stand for movies where two rings trade places
by the ring of the under strand flying through the ring of the  over strand. The dotted end represents a tube ``capped off'' at the bottom
by a disk. Generators 4 and 5 stand for singular ``foam vertices'', and will be referred to as the positive and negative vertex, respectively.
The positive vertex represents the movie shown on the left: the right ring approaches the left ring from below, flies inside it and merges with it.
The negative vertex represents a ring splitting and the inner ring flying out below and to the right. To be 
completely precise, $\wTFe$ as a circuit algebra has more vertex generators than shown above: the vertices appear with all possible orientations of the strands.
However, all other versions can be obtained from the ones shown above using ``orientation switch'' operations (to be discussed in Section \ref{subsec:wops}).


\parpic[r]{\input figs/MixedCrossings.pstex_t }
The thin red strands denote one dimensional strings in $\bbR^4$, or ``flying points in $\bbR^3$''. The crossings between the two types of strands 
(generators 6 and 7) represent ``points flying through rings''. For example, the picture on the left shows generator 6, where ``the point on the right approaches the ring on the left from below, flies
through the ring and out to the left above it''.
This explains why there are no generators with a thick strand crossing under a 
thin red strand: a ring cannot fly through a point. 

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

\begin{figure}
 \input figs/StringOnTube.pstex_t 
 \caption{A string-tube vertex.}\label{fig:MixedVertex}
\end{figure}

%``Outside wall'' makes no sense but there is a framing issue where the string can wrap around the tube a number of times before it attaches itself.
% Since we have introduced only one kind of string to tube vertex, puncture is restricted later.
% Wens commute with string to tube vertices and disappear when punctured.


\subsection{The relations}\label{subsec:wrels}
As a list, the relations for $\wTFe$ are the same as the relations for $\wTF$ \cite[Section 4.5]{Bar-NatanDancso:WKO2}: 
$\{$\Rs, R2, R3, R4, OC, CP$\}$. 
Recall that \Rs~is the weak (framed) version
of the Reidemeister 1 move; R2 and R3 are the usual second and third Reidemeister moves; R4 allowes moving a strand over or under a vertex. 
OC stands for {\em Overcorssings Commute}, CP for {\em Cap Pullout}: these
two relations are shown in Figure~\ref{fig:wTFeRels}, for a detailed explanation see \cite[Section 4.1.2]{Bar-NatanDancso:WKO2}.
\begin{figure}
 \input figs/CapRel.pstex_t 
 \caption{The OC and CP relations.}\label{fig:wTFeRels}
\end{figure}

In $\wTFe$ all relations should be interpreted in all
possible combinations of strand types and orientations (tube or string), for example the lower strand of the R2 relation can either be thick black or thin red, as shown below:
\begin{center}
 \input figs/R2.pstex_t
\end{center}
Similarly, any of the lower strands of the R3, R4, and OC relations may be thin red.

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

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







\subsection{The operations}\label{subsec:wops}
Like $\wTF$, $\wTFe$ is equipped with a set of auxiliary operations in addition to the circuit algebra structure.

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

Topologically, the operation ``orientation switch'', denoted 
$S_e$ for a given strand $e$, acts by reversing both the (1-dimensional) direction and the (2-dimensional) orientation of the tube $e$. 
Diagrammatically, this corresponds to simply reversing the direction of the corresponding strand $e$. 

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


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

A related operation, {\it disc unzip}, is unzip done on a capped strand, pushing the tube off in the direction of the framing 
(in diagram world, in the direction of the blackboard framing), as 
before. 
An example is shown in Figure~\ref{fig:DiscUnzip}; see \cite[Section 4.1.3]{Bar-NatanDancso:WKO2} for details on framings and unzips.
\begin{figure}
\input{figs/StringUnzip.pstex_t} \quad \quad \quad \quad \input{figs/CapUnzip.pstex_t}
\caption{Unzip and disc unzip.}\label{fig:DiscUnzip}
\end{figure}

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

\begin{figure}
\input figs/PunctureRule.pstex_t 
\caption{Puncture operations: the picture on the left shows which edges can be punctured at each vertex. The middle and right pictures show the effect of puncture operations.}\label{fig:punctures}
\end{figure}

For simplicity, we place a restriction on which strands can be punctured, namely at each (fully thick black) vertex puctures are only allowed for 
one of the three meeting strands, as shown on the left of Figure \ref{fig:punctures}. More general puctures could be allowed in a theory 
with more than one kind of ``string to tube'' vertex. The right of the same figure
shows that when puncturing one of the thick strands of a mixed vertex, the puncture ``spreads''. Topologically, this is because the mixed
vertex represents a string attached to a tube, so when puncturing $e$, the entire tube retracts to its core.
Finally, a capped tube disappears (deformation retracts to a point) when punctured.

In summary,
\[
  \wTFe=\CA\!\left.\left.\left\langle
  \raisebox{-8mm}{\input{figs/wTFgens.pstex_t} }
  \right|
  \parbox{0.9in}{\centering \Rs, R2, R3, R4, OC, CP \\ relations }
  \right|
  \parbox{0.8in}{\centering $S_e, A_e,$ $u_e, d_e, p_e$ \\ auxiliary operations}
  \right\rangle
\]


 






\subsection{The associated graded structure $\calA^{sw}$}
As in \wko, the space $\wTFe$ is filtered by powers of the augmentation ideal and its associated graded circuit algebra, 
denoted $\calA^{sw}$, is a ``space
of arrow diagrams on foam skeletons with strings''. As a circuit algebra, $\calA^{sw}$ is presented as follows:
\[
  \calA^{sw}=\CA\!\left.\left.\left\langle
  \raisebox{-8mm}{\input{figs/AswGens.pstex_t}}
  \right|
  \parbox{0.8in}{\centering relations as below}
  \right|
  \parbox{0.8in}{\centering auxiliary operations as below}
  \right\rangle.
\]
Generators 1 and 5 are called single arrows and they are of degree one, while all others are ``skeleton features'' of degree zero.
The relations are almost the same as in \cite[Section 4.2.1]{Bar-NatanDancso:WKO2}, which describes the relations for 
the associated graded of $\wTF$: 
$\aft$ (the 4-Term relation), TC (Tails Commute), RI (Rotation 
Invariance), CP (the arrow Cap Pullout), and VI (Vertex Invariance). For $\wTFe$ there is an 
additional relation TF (Tails Forbidden on strings).
The TC and $\aft$ relations are shown in Figure \ref{fig:TCand4T}. The Vertex Invariance relation is shown in Figure \ref{fig:VI}: here the $\pm$ signs depend on the 
strand orientations. Note that the type of the 
vertex and the types of each strand (thick black or thin red) are left undetermined: the VI relation applies in all cases. Figure \ref{fig:RICPTF} shows
the other relations: RI, CP and TF. Note that technically TF is not a relation: there were no generators with an arrow tail on a thin red strand,
so saying that such an element vanishes is meaningless. However, without TF the VI relation would have to be stated for all the sub-cases of 0, 1 or 3 thin red
strands, so we prefer this cleaner way, even if it is a  
slight abuse of notation.

\begin{figure}
 \input{figs/TCand4T.pstex_t}
 \caption{The TC and $\protect\aft$ relations. Note that the 3rd strand in each term of the $\protect\aft$ relation can be 
either thick black or thin red, the relation applies in either case.} 
 \label{fig:TCand4T}
\end{figure}

\begin{figure}
 \input{figs/VI.pstex_t}
 \caption{The VI relation: the vertices and strands could be of any type, but the same throughout the relation.}
 \label{fig:VI}
\end{figure}
\begin{figure}
 \input{figs/RICPTF.pstex_t}
 \caption{The RI and CP relations, and the TF relation (which is not really a relation).}
 \label{fig:RICPTF}
\end{figure}

Denote arrow diagrams on a given skeleton $S$ by $\calA^{sw}(S)$. In particular, $\calA^{sw}(\uparrow_n)$ denotes arrow diagrams on $n$ (black) vertical strands, and $\calA(\downcap_n)$ denotes arrow diagrams on $n$ capped strands. 

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

To summarise:
\[
  \calA^{sw}=\CA\!\left.\left.\left\langle
  \raisebox{-8mm}{\input{figs/AswGens.pstex_t}}
  \right|
  \parbox{1in}{\centering $\aft$, TC, VI, CP, RI, TF \\ relations}
  \right|
  \parbox{1.1in}{\centering $S_e, A_e, u_e,$ $d_e, p_e$ \\auxiliary operations}
  \right\rangle
\]

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

\begin{theorem}
 The natural inclusion of diagrams induces a circuit algebra isomorphism $\calA^{sw}\cong \calA ^{swt}$. Furthermore,
 the $\aAS$ and $\aIHX$ relations of Figure \ref{fig:ASIHXSTU} hold in $\calA^{swt}$.
\end{theorem}

\begin{figure}
 \input{figs/ASIHXSTU.pstex_t}
 \caption{The $\protect\aAS$, $\protect\aIHX$ and the three $\protect\aSTU$ rerations. Note that in $\protect\aSTU_1$, the skeleton strand can be thin red
 or thick black, and that $\protect\aSTU_3$ is the same as the TC relation.}
 \label{fig:ASIHXSTU}
\end{figure}

The proof is identical to the proof of \cite[Theorem 3.8]{Bar-NatanDancso:WKO2}. In light of this isomorphism, we will drop the extra ``$t$'' from the
notation and use $\calA^{sw}$ to denote either of these spaces. 

The space $\calA^{sw}(\uparrow_n)$ forms a Hopf algebra with the stacking product and the standard co-product. (The coproduct is the sum over all possible ways of distributing the connected components of the arrow graph between two copies of the skeleton.) 
As in \cite{Bar-NatanDancso:WKO2}, the primitive elements of $\calA^{sw}(\uparrow_n)$ are connected diagrams,
denoted $\calP^{sw}(\uparrow_n)$, and $\calP^{sw}(\uparrow_n)=\langle\text{trees}\rangle\oplus\langle\text{wheels}\rangle$ as a vector space. Examples of trees and wheels are shown in Figure \ref{fig:TreeAndWheel}; 
for details
see \cite[Section 3.1]{Bar-NatanDancso:WKO2}. Note that the RI relation can now be rephrased (via $\aSTU_2$) as the vanishing of the wheel with a single spoke, or one-wheel.
\begin{figure}
 \input{figs/TreeAndWheel.pstex_t}
 \caption{An example of a {\em tree}, left, and a {\em wheel}, right.}
 \label{fig:TreeAndWheel}
\end{figure}



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

\begin{fact}\label{fact:semidirect}
As a Lie algebra, $\calP^{sw}(\uparrow_n)\cong \langle \text{wheels}\rangle \rtimes \langle \text{trees} \rangle$. The vector space (abelian Lie algebra) spanned by wheels is canonically isomorphic to the space $(\cyc_n)_{\geq 1}$ of cyclic words\footnote{Cyclic words are denoted $\mathfrak{tr}_n$ in \cite{Bar-NatanDancso:WKO2} and \cite{AlekseevTorossian:KashiwaraVergne} and $\mathfrak{T}_n$ in \cite{AlekseevEnriquezTorossian:ExplicitSolutions}.} in $n$ letters of degree at least 2, where degree is given by word length, and degree 1 is killed by the RI relation. 
\end{fact}



\begin{fact}\label{fact:CapIsWheels}
 $\calA^{sw}(\downcap)$, the part of $\calA^{sw}$ with skeleton a single capped strand, is isomorphic as a vector space to the completed polynomial algebra
 freely generated by wheels $w_k$ with $k \geq 2$.
\end{fact}

\begin{fact}\label{fact:VTwoStrands}
 $\calA^{sw}(\raisebox{-1mm}{\input{figs/PlusVertex.pstex_t}})\cong \calA^{sw}(\uparrow_2)$, where $\calA^{sw}(\raisebox{-1mm}{\input{figs/PlusVertex.pstex_t}})$ 
 stands for the space of arrow diagrams whose skeleton is a single vertex (the picture shows a positive vertex but the statement is 
 true for all kinds of vertices with thick black strands), 
 and $\calA^{sw}(\uparrow_2)$ is the space of arrow diagrams on two (thick black) strands. 
\end{fact}

The following Lemma -- called the {\em Sorting Lemma} as we will see it ``sorts'' arrow tails above arrow heads -- will play an important role. In particular the second isomorphism stated is the map $\varphi$ appearing in Theorem~\ref{thm:main}, part $(2)$. We will refer to the isomorphism $\varphi$ in the Lemma as the {\em sorting isomorphism}.

\begin{lemma}[Sorting Lemma]\label{lem:CapString}
There is a linear isomorphism $\varphi: \calA^{sw}\Big(\raisebox{-2.3mm}{\input{figs/LinesWithStrings.pstex_t}}\Big)\stackrel{\cong}{\longrightarrow} \calA^{sw}(\uparrow)$ between the vector spaces of arrow diagrams on the indicated skeleta. On the left, the thin red string is a tangle end. The black strand may continue past the arrow, and there may be additional skeleton components: the same on both sides.  
 Applying the isomorphism $\varphi$ twice, one obtains 
 $\calA^{sw}\Big(\raisebox{-2.3mm}{\input{figs/VtxWithStrings.pstex_t}}\Big)\stackrel{\varphi}{\cong} \calA^{sw}(\uparrow_2)$.
 \end{lemma} 



\begin{proof}
 We construct inverse maps between the two spaces. There is a natural map 
 $\calA^{sw}(\uparrow) \stackrel{\psi}{\rightarrow} \calA^{sw}\Big(\raisebox{-2.3mm}{\input{figs/LinesWithStrings.pstex_t}}\Big)$, shown in Figure~\ref{fig:SlideUpLemma}: given an arrow diagram on a single thick black strand, place all arrow endings (denoted ``$x$") on the strand above the tube/string vertex.
  

 \begin{figure}
 \input figs/SlideUpLemma.pstex_t
 \caption{Inverse maps.}
 \label{fig:SlideUpLemma}
 \end{figure}

  
 In the other direction, consider an arrow diagram on the capped/stringed vertex. One may
 assume that there are only arrow tails on the capped strand under the vertex: any arrow head may be commuted using $\aSTU$ relations towards the cap, where it is killed by the CP relation\footnote{This argument also appears in \cite{Bar-NatanDancso:WKO2}, for example as the basic idea for the proof of Fact~\ref{fact:CapIsWheels}.}. On the thin red strand there are only arrow heads. 
 To construct  $\varphi$, first ``push" the arrow tails (denoted ``$t$") from the capped strand up across the
 vertex using the VI relation. Since tails vanish on the thin red strand, they simply slide past
 the vertex. Once the capped side is cleared, continue by sliding the arrow heads ``$h$" up from the thin red string to the strand above the vertex.
 Now the cap relation kills any arrow heads on the capped strand, so once again they simply slide past the vertex. The result placed on a single thick black strand is shown in Figure~\ref{fig:SlideUpLemma}. 
  
 
 It is clear that $\psi$ is well-defined, we leave it to the reader to check that so is $\varphi$ as a short exercise. Given that both maps are well-defined, it is clear that they are inverses of each other. \qed
 \end{proof}
 
 
Observe that in the image of $\varphi$, all arrow tails are above arrow heads along the strand. Arrow diagrams of this form appear in the context of ``over-then-under'' tangles, which have applications in several contexts, including virtual braid classification \cite{BDV:OU}.





\subsection{The homomorphic expansion}

As discussed in \cite[Section 2.3]{Bar-NatanDancso:WKO2}, an expansion for $\wTFe$ is a map $Z^w: \wTFe \to \calA^{sw}$ with the property that the associated graded map \linebreak
$\operatorname{gr} Z^w: \calA^{sw} \to \calA^{sw}$ is the identity map $\operatorname{id}_{\calA^{sw}}$. A homomorphic expansion is an expansion which also
intertwines each operation of $\wTFe$ with its arrow diagrammatic counterpart. In 
\cite[Theorems 4.9 and 4.11]{Bar-NatanDancso:WKO2} we proved that the existence of solutions for the Kashiwara--Vergne equations implies that
there exsists a homomorphic expansion for $\wTF$. In fact that homomorphic expansions\footnote{Subject to the minor technical condition that the value of
the vertex doesn't contain isolated arrows.} for $\wTF$  are in one-to-one correspondence with solutions
to the Kahiwara-Vergne problem. 

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

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

To state the full characterisation, we use co-simplicial notation in subscripts. For example, for $R=e^{a_{12}}=\in \calA^{sw}(\uparrow_2)$, $R_{13}=e^{a_{13}}$ and $R_{23}=e^{a_{23}}$ in $\calA^{sw}(\uparrow_3)$ are the diagrams where $R$ is placed on strands 1 and 3, and 2 and 3, respectively. $R_{(12)3} \in \calA^{sw}(\uparrow_3)$ is obtained by doubling the first strand of $R$ and placing it on strands 1 and 2, and placing the second strand of $R$ on strand 3, that is, $R_{(12)3}=e^{a_{13}+a_{23}}$. Similarly for $V\in \calA(\uparrow_2)$, $V_{12}\in \calA(\uparrow_ 3)$ denotes $V$ placed on the first two starnds, et cetera.


\begin{fact} \label{fact:EquationsForZ}
A filtered, group-like map ${Z}^{ow}: \wTF \to {\calA}^{osw}$ is a homomorphic expansion if and only if the $Z^{ow}$-values $V=Z^{ow}(\pvertex)$ and $C=Z^{ow}(\downcap)$ satisfy the following equations:

\begin{enumerate}
\item  {\em R4 Equation}:
\begin{equation}\tag{R4}\label{eq:R4}
V_{12}R_{(12)3}=R_{23}R_{13}V_{12}   \quad \text{ in } \calA^{sw}(\uparrow_3).
\end{equation}
\item {\em Unitarity Equation:}
\begin{equation}\tag{U}\label{eq:U}
V\cdot A_1A_2(V)=1   \quad \text{ in } \calA^{sw}(\uparrow_2),
\end{equation}
where $A_1$ and $A_2$ denote the antipode operations.

\item {\em Cap Equation\footnote{For convenience we state the Cap Equation phrased for caps at the bottom of strands, hence the difference from the equivalent formulation in \cite{Bar-NatanDancso:WKO2}.}:}
\begin{equation}\tag{C}\label{eq:C}
C_{(12)}V^{-1}_{12}=C_1C_2   \quad \text{ in } \calA^{sw}(\downcap_2),
\end{equation}
where the subscripts mean strand placements as in the R4 Equation.
\end{enumerate}
\end{fact}





We begin by showing that finding a homomorphic expansion for $\wTFe$ is no harder than finding one for $\wTF$.


\begin{theorem}\label{thm:ExtendRestrict}
Homomorphic expansions for $\wTF$ are in one-to-one correspondence with homomorphic expansions for $\wTFe$ via unique extension and restriction.
\end{theorem}

\parpic[r]{
\xymatrix{
 \wTF  \ar@{^{(}->}[r]  \ar[d]^{{Z}^{ow}} 
 & \wTFe  \ar[d]^{{Z}^w} \\
 {\calA}^{osw}  \ar@{^{(}->}[r]
 & {\calA}^{sw}
 }
}
{\em Proof.} Every element of $\wTFo$ is also in $\wTFe$, hence any $Z^w$ restricts to a homomorphic expansion $Z^{ow}$ of $\wTFo$. Every element of $\wTFe$ is the result of puncturing -- possibly on multiple strands -- an element of $\wTFo$, and $Z^w$ is required to commute with punctures. Hence any $Z^{ow}$ uniquely extends to a $Z^w$.  \qed


\vspace{2mm}


In \cite[Section 4.4]{Bar-NatanDancso:WKO2} we showed that short arrows -- arrows whose head and tail is not separated by any other arrow endings -- supported on either strand of $V$ don't affect whether $Z^w$ 
 is a homomorphic expansion. That is, if $Z^w$ is a homomorphic expansion and $a$ is a linear combination of short arrows, then
 replacing $V$ by $e^a V$ gives rise to another homomorphic expansion. Hence, in \cite{Bar-NatanDancso:WKO2} we typically assume there are no short arrows in $V$, this motivates the following definition:
 
 \begin{definition}
 A homomorphic expansion $Z$ is {\em v-small} if there are no short arrows in the $Z$-value $V$ of the positive vertex.
 \end{definition}

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

\begin{lemma}\label{lem:pV} For any v-small homomorphic expansion $Z^w$, $Z^w\Big(\raisebox{-2mm}{\input{figs/PunctVertex.pstex_t}}\Big)=1$, 
that is, the $Z^w$-value of a left punctured vertex is 
trivial.
\end{lemma}

\begin{proof}
 Recall from \cite[Proof of Theorem 4.9]{Bar-NatanDancso:WKO2} that the $Z^w$-value $V$ of the positive (not punctured) vertex can be written as
 $V=e^be^{t}$, where $b$ is a linear combination of wheels only and $t$ (denoted $uD$ in \cite{Bar-NatanDancso:WKO2}) is a linear combination of trees. Puncturing the left strand
 of $V$ kills all arrow diagrams with tails on the left strand. Diagrams that survive are wheels, and trees all of whose tails are on the right side strand. However, if all tails of a tree are supported on one strand, then the tree
 is a single arrow, due to TC and the anti-symmetry of the trivalent arrow vertices, thus the only surviving trees are simple arrows directed from right to left. Observe that all of these arrow diagrams commute with each other in $\calA^{sw}(\uparrow_2)$.

 Denote the value of the punctured vertex by $p_1V=e^{p_1(b)}e^{p_1(t)}$. Recall that $V$ must satisfy the Unitarity Equation of Fact~\ref{fact:EquationsForZ},
 so $p_1V\cdot A_1A_2(p_1 V)=1$. Since wheels have only tails, $A_1A_2(p_1(b))=p_1(b)$. Each arrow has one head, 
 so $A_1A_2(p_1(t))=-p_1(t)$. Hence, using commutativity, $p_1V\cdot A_1A_2(p_1 V)=e^{2p_1(b)}=1$, which implies that $p_1(b)=0$.
 As for $p_1(t)$, one can show that there are no arrows pointing from the right to the left strand by a direct computation in degree 1.
 \qed
 \end{proof}




%--------------------------------------------------------------------------------------------------------





\section{Proof of Theorem \ref{thm:main}}\label{sec:Proof}

\subsection{Proof of Theorem~\ref{thm:main} Part (1)}\label{subsec:Part1}
We prove Part 1 in two steps: first verifying the easier ``tree level" case, which nonetheless contains the main idea, then in general.

\subsubsection{Tree level proof of Part (1).}\label{subsec:Part1TreeProof}
Let $\calA^{tree}$ denote the quotient of $\calA^{sw}$ by all wheels, and let $\pi:\calA^{sw} \to \calA^{tree}$ denote the quotient 
map (cf \cite[Section 3.2]{Bar-NatanDancso:WKO2}). Part (1) of the main theorem is the same as stating that $Z^w$ is determined by $Z^u$. $Z^w$, in turn is determined by the values 
$V$ and $C$ of the positive vertex and the cap \cite[Sections 4.3 and 4.5]{Bar-NatanDancso:WKO2}, so one only needs to show that $V$ and $C$ are
determined by $Z^u$. Proving this ``on the tree level'' means showing only that $\pi(V)$ and $\pi(C)$ are determined by $Z^u$. In particular, observe that since $C$ is a linear combination of products of wheels
(Fact \ref{fact:CapIsWheels}), we have $\pi(C)=1$, so we only need to show that $\pi(V)$ is determined by $Z^u$.

\parpic[r]{\input{figs/Buckle.pstex_t}}
Let $B^u$ denote the ``buckle'' $\sKTG$, as shown on the right (ignore the dotted lines for now). All edges are oriented up, and by the drawing conventions
of \cite[Section 4.6]{Bar-NatanDancso:WKO2} all the vertices in the bottom half of the picture are negative and all the ones in the top half are positive. 
Let $B^w=a(B^u)\in \wTFe$, and $\beta^u:=Z^u(B^u)$. Note that $\beta^u$ can be represented as a chord diagram on four strands\footnote{The value of $\beta^u$ is computed explicitly to degree four in \cite{Bar-Natan:WKO4}.}: use VI relations to move all chord endings
to the ``middle'' of the skeleton, between the dotted lines on the picture. Hence, we write $\beta^u \in \calA^u(\uparrow_4)$. 
Let $\beta^w=\alpha(\beta^u)$, and note that by the compatibility of $Z^u$ and $Z^w$
we have $\beta^w=Z^w(B^w)$. We will perform a series of operations on $B^w$ and $\pi(\beta^w)$ to recover $\pi(V)$ from it.

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


\begin{figure}[h]
 {\input{figs/BuckleToV.pstex_t}}
 \caption{From the ``buckle'' $\beta^w$ to the (modified) vertex.}
 \label{fig:BuckleToV}
\end{figure}

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

On the other hand, note that the space of chord diagrams on the skeleton of $K$ is the space $\calA(\uparrow_2)$ by the Sorting Lemma (Lemma \ref{lem:CapString}) and VI. 
Note also that $K$ is a circuit algebra combination of a vertex, two left-punctured right-capped vertices and an all-red-strings vertex, and the $Z^w$-values of the latter three are trivial. So $\pi(Z^w(K))=\pi(V) \in \calA^{tree}(\uparrow_2)$. Hence, $\pi(V)$ is determined by $Z^u$ as needed. \qed


%----------------------------------------------------------------------------------------------------------



\subsubsection{Complete proof of Theorem~\ref{thm:main} Part (1)}\label{subsec:Part1Proof}
In the previous subsection we showed that $Z^u$ determines $\pi(V)\in \calA^{tree}(\uparrow_2)$. The proof of Part (1) is completed by the following Lemma:

\begin{lemma}\label{lem:PiVEnough} For any homomorphic expansion $Z^w$ of $\wTFe$ set $V=Z^w(\pvertex)$ and $C=Z^w(\downcap)$. Then
$\pi(V)\in \calA^{tree}(\uparrow_2)$ determines both $V$ and $C$ uniquely.
\end{lemma}

\begin{proof}
We use a perturbative argument. By contradiction, assume this is not the case, in particular, first assume that there exist $V\neq V'$, both of which are vertex values of $Z^u$-compatible homomorphic expansions, such that $\pi(V)=\pi(V')$. Let $v$ denote the lowest degree term of $V-V'$. Note that $v$ is primitive and $v\in \ker \pi$, so $v$ is a homogeneous linear combination of wheels. By the Unitarity Equation of Fact~\ref{fact:EquationsForZ}, we have $A_1A_2(v)=-v$. Recall that $A_i$ reverses the direction of the strand $i$ and multiplies each arrow diagram by $(-1)$ to the number of heads on that strand. Since $v$ has only tails, $A_1A_2(v)=v$, so $v=-v$, so $v=0$, a contradiction. Therefore, $\pi(V)$ determines $V$ uniquely.

Now we show that $V$ determines $C$ uniquely. Assume there are different values $C$ and $C'$ in $\calA^{sw}(\downcap)$ so that $(V,C)$ and $(V,C')$ are both vertex-cap value pairs of $Z^u$-compatible homomorphic expansions. Let $c$ denote the lowest degree term of $C-C'$, then $c$ is a scalar multiple of a single wheel. The Cap Equation of Fact~\ref{fact:EquationsForZ}
implies $c_{(12)}=c_1+c_2$ in $\calA^{sw}(\downcap_2)$. 

There is a well-defined linear map $\omega: \calA^{sw}(\downcap_2) \to \bbQ[x,y]$ sending an arrow diagram -- which has arrow tails only on each strand -- to ``$x$ to the power of the number of tails on strand 1, times $y$ to the power of the number of tails on strand 2''. Assume $c= \alpha w_r$, where $w_r$ denotes the $r$-wheel, and $\alpha \in \bbQ$. Then $0=\omega(c_{(12)}-c_1-c_2)=\alpha((x+y)^r-x^r-y^r)$,
so either $r=1$ or $\alpha=0$. But $w_1=0$ in $\calA^{sw}$ by the $RI$ relation, hence $\alpha=0$ and thus $c=0$, a contradiction.  
\qed
\end{proof}



%-----------------------------------------------------------------------------------------------------------



\subsection{Proof of Theorem~\ref{thm:main} Part (2)}\label{subsec:AETFormula}
In this section we compute $V$, the value of the vertex, from $\Phi$, the Drinfel'd associator determining $Z^b$, using the construction of Part~(1). In Appendix~\ref{app:AET} we also show that this result
translates to the \cite{AlekseevEnriquezTorossian:ExplicitSolutions} formula for Kashiwara-Vergne solutions in terms of Drinfel'd associators.

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

\begin{lemma}\label{lem:PhiFacts} Let $\Phi=\Phi(c_{12},c_{23})\in \calA^u(\uparrow_3)$ be a Drinfel'd associator, where $c_{ij}$ denotes a chord between strands $i$ and $j$. Let $\alpha(\Phi)\in \calA^w(\uparrow_3)$ denote the image of $\Phi$ in arrow diagrams. Then, the following facts hold:
\begin{enumerate}
\item \label{it:Central} $\Phi(x,y)=\Phi(x,-x-z)$, whenever $(x+y+z)$ is central.
\item \label{it:TwoPunctures} $p_ip_j\alpha(\Phi)=1$, whenever $i,j\in \{1,2,3\}$, $i\neq j$, and $p_i$ denotes puncture of the $i$-th strand.
\item \label{it:Concat} $\mu_{23}\left(S_2p_1(\alpha(\Phi))\right)=1$, where  $S_2$ stands for orientation switch of strand 2, and $\mu_{23}$ is concatenation (multiplication) of strands 2 and 3 -- a circuit algebra operation -- as shown in Figure~\ref{fig:Concat}.
\end{enumerate}
\end{lemma}

\begin{figure}
\input figs/concat.pstex_t
\caption{A concatenated associator.}\label{fig:Concat}
\end{figure}

\begin{proof}
Property~\eqref{it:Central} follows from the fact that the logarithm of $\Phi$ is a Lie series in $x$ and $y$ with no constant term.

To show Property \eqref{it:TwoPunctures}, recall that if $\Phi(x,y)$ is a Drinfel'd associator, then $\Phi(0,y)=\Phi(x,0)=1$.
Therefore $p_1p_2\alpha(\Phi)=1$, because $p_1p_2\alpha(c_{12})=p_1p_2(a_{12}+a_{21})=0$. Similar reasoning shows that $p_2p_3\alpha(\Phi)=1$. 
Finally, $p_1p_3\alpha(\Phi(c_{12},c_{23}))=\Phi(a_{21},a_{23})$, and since $[a_{21},a_{23}]=0$ by the TC relation, $\Phi(a_{21},a_{23})=1$.  

For Property~\ref{it:Concat}, note that $p_1(\alpha(\Phi))=\Phi(a_{21},-a_{21}-a_{31})$. Thus, strands 2 and 3 carry only arrow tails, and these commute by the TC relation, and $S_2p_1(\alpha(\Phi))=\Phi(-a_{21},a_{21}-a_{31})$. Furthermore, tails on strand 3 can be pulled to strand 2 through the concatenation, which identifies $a_{21}$ with $a_{31}$. Therefore, $\Phi(-a_{21},a_{21}-a_{31})=\Phi(-a_{21},0)=1$.  \qed
\end{proof}

\medskip

To compute $V$ and prove Part (2) of Theorem~\ref{thm:main}, consider once again the w-tangled foam $K$ on the right of Figure~\ref{fig:BuckleToV}. 

On one hand, $Z^w(K)$ can be computed directly from the generators: $Z^w(K)=C_1C_2V_{12}\in \calA^{sw}(\uparrow_2)$, since the values of the left-punctured vertices are trivial. Hence, if we know $Z^w(K)$, we know $V$.

On the other hand, we can compute $Z^w(K)$, using the compatibility with $Z^u$, as follows. Note that $B^u$ is the closure -- in the sense of \eqref{eq:cl} -- of the parenthesised braid $B^{b}$ shown
in Figure~\ref{fig:BuckleBraid}, $B^w=a(B^u)$. Using the notation
$\beta^u=Z^u(B^u)$, and $\beta^w=Z^w(B^w)$, and by the compatibility of $Z^w$ with $Z^u$, we have $$\beta^w=Z^w(B^w)=\alpha(Z^u(B^u))=\alpha(\beta^u).$$
How does $Z^w(K)$ differ from $\beta^w$? To obtain $K$, a vertex and a cap were attached to $B^w$, two strands were punctured and the cap unzipped, as in Figure~\ref{fig:BuckleToV}. The $Z^w$-value of the added vertex cancels when its left strand is punctured, however, the value of the cap remains and is unzipped. Thus, in loose notation, $Z^w(K)= u(C)\cdot p^2(\beta^w)$, where $p^2$ denotes the two punctures -- we will compute this value explicitly in terms of associators shortly. 

To equate the two approaches, we need to express $u(C)\cdot p^2(\beta^w)$ as an element of $\calA^{sw}(\uparrow_2)$, by applying the {\em sorting isomorphism} $\varphi$ of Lemma~\ref{lem:CapString}. By doing so, we obtain  
\begin{equation}\label{eq:BuckleV}
C_1C_2V_{12}=\varphi(u(C)p^2(\beta^w)).
\end{equation}
Through a careful analysis of the right hand side, this will imply formula (\ref{eqn:AET}) stated in Theorem~\ref{thm:main}. In other words, we need to compute
$$\Upsilon:=\varphi(u(C)p^2(\beta^w)).$$

To achieve this, we use that $\beta^w=\alpha(\beta^u)$, and compute $\beta^u$ in terms of the Drinfel'd associator $\Phi$ associated to $Z^u$.
By the compatibility of $Z^u$ and $Z^b$, it is enough to compute $\beta^b:=Z^b(B^b)$. The result can be read 
from the picture in Figure~\ref{fig:BuckleBraid}:
$$\beta^b=\Phi^{-1}_{(13)24}\Phi_{132}R_{32}\Phi^{-1}_{123}\Phi_{(12)34}.$$
Recall that the cosimplicial notation used in the subscripts show which strands
the diagrams are placed on, for example, $\Phi^{-1}_{(13)24}=\Phi^{-1}(c_{12}+c_{32}, c_{24})$. 
Also recall that $R=e^{c/2}$, so $R_{32}=e^{c_{23}/2}$. 

\begin{figure}
 \input{figs/BuckleBraid.pstex_t}
 \caption{Computing $\beta^b$. Strands are numbered at the top and multiplication is read from bottom to top; the rightmost column lists the images of the factors under $p_1p_3\alpha$.}\label{fig:BuckleBraid}
\end{figure}

As $\beta^u$ is the tree closure of $\beta^b$, it is given by the same formula interpreted as an element of $\calA^u(\uparrow_4)$. 
One then applies $\alpha$ to obtain $\beta^w=\alpha(\beta^u)$. After the vertex and cap attachment, of Figure~\ref{fig:BuckleToV},
strands 1 and 3 are punctured and strands 2 and 4 are capped, and in this strand numbering, $u(C)=C_{24}$ Therefore, we have
$$\Upsilon=\varphi\left(C_{24}\cdot p_1p_3\alpha(\Phi^{-1}_{(13)24}\Phi_{132}R_{32}\Phi^{-1}_{123}\Phi_{(12)34})\right).$$



Next, we analyse how the puctures and $\alpha$ act on factors of $\beta^b$.
First observe that $p_3\alpha(R_{32})=e^{a_{23}/2}$, where $a_{ij}$ is a single arrow pointing from strand $i$ to strand $j$.

Observe that $p_1p_3\alpha(\Phi^{-1}_{123})=p_1p_3\alpha(\Phi^{-1}_{123})=1$ by Fact \eqref{it:TwoPunctures} of Lemma~\ref{lem:PhiFacts}.

Since strands 1 and 3 are both punctured, no arrows 
can be supported between these two strands, hence $p_1p_3\alpha(\Phi_{(12)34})=\Phi(a_{23},a_{43})$. 


By Properrty~\eqref{it:Central} of Lemma~\ref{lem:PhiFacts},
$\Phi^{-1}_{(13)24}=\Phi^{-1}(c_{(13)2},c_{24})=
\Phi^{-1}(c_{(13)2},-c_{(13)2}-c_{(13)4})$, so $p_1 p_3 \alpha \Phi^{-1}_{(13)24}= \Phi^{-1}(a_{2(13)},-a_{2(13)}-a_{4(13)})$.
To summarise, 
$$
\Upsilon= \varphi\left(C_{24}\cdot \Phi^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)}) \cdot e^{a_{23}/2} \cdot \Phi(a_{23},a_{43})\right).
$$
 
Note that the expression $\Phi^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)}) \cdot e^{a_{23}/2} \cdot \Phi(a_{23},a_{43})$ has only arrow tails on strands 2 and 4, and therefore commutes with $C_{24}$ by the $TC$ relation. Hence, by the definition of $\varphi$,
\begin{flalign*}
\Upsilon &= \varphi\left(\Phi^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)}) \cdot e^{a_{23}/2} \cdot \Phi(a_{23},a_{43})\cdot C_{24}\right) \\ 
&= \varphi\left(\Phi^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)}) \cdot e^{a_{23}/2} \cdot \Phi(a_{23},a_{43})\right)\cdot\varphi(C_{24}).
\end{flalign*} 
Furthermore, by the strand numbering convention shown in Figure~\ref{fig:NumberingK}, we have  $\varphi(C_{24})=C_{12}$.
Therefore, 
$$V_{12}=C_1^{-1}C_{2}^{-1}\Upsilon= C_1^{-1}C_2^{-1}\varphi\left(\Phi^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)}) \cdot e^{a_{23}/2} \cdot \Phi(a_{23},a_{43})\right)C_{12},$$
as stated in part (2) of Theorem~\ref{thm:main}. \qed

\begin{figure}
\input{figs/NumberingK.pstex_t}
\caption{Strand numbering convention for $K$ and $V$: arrow endings from strand 1 and 2 of $K$ are ``pushed'' to strand 1 of $V$ when applying $\varphi$, and arrow endings from strands 3 and 4 are pushed to strand 2.}\label{fig:NumberingK}
\end{figure}

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

 
 
%-----------------------------------------------------------------------------------------
 
 



\subsection{Proof of Theorem~\ref{thm:main} part (3): the double tree construction.}\label{subsec:DT}
Given a homomorphic expansion $Z^u$ of $\sKTG$, in Section~\ref{subsec:Part1Proof} we showed that if there is to exist a homomorphic expansion $Z^w$ of $\wTFe$ compatible with $Z^u$, then $V=Z^w(\pvertex)$ and $C=Z^w(\downcap)$, and hence $Z^w$ itself, are uniquely determined by $Z^u$. In Section~\ref{subsec:AETFormula} we proved the formula \eqref{eqn:AET}, which in particular gives an explicit expression for the tree level value of $V$ in terms of the Drinfel'd associator associated to $Z^u$. From here on we denote this value $V^{tree}_\beta$ as it is calculated from the $Z^u$-value of the ``buckle'' graph:
\begin{equation}\label{eq:VtreeAET} 
V^{tree}_\beta=\varphi\left(\Phi^{-1}(a_{2(13)},-a_{2(13)}-a_{4(13)})e^{a_{23}/2} \Phi(a_{23},a_{43}) \right)
\end{equation}


It remains to show that there exists an appropriate cap value $C=Z^w(\downcap)$ such that $C$ along with the value of the vertex as given in formula ~\eqref{eqn:AET} 
\begin{equation}\label{eq:VBeta}
Z^w(\pvertex)=:V_\beta= C_1^{-1}C_2^{-1}V^{tree}_\beta C_{(12)}
\end{equation}
determine a homomorphic expansion of $\wTFe$, which is furthermore compatible with $Z^u$. In particular, in order to show that the a pair $(V_\beta,C)$ defines a homomorphic expansion, one needs to show that the values $V_\beta$ and $C$ satisfy the equations \eqref{eq:R4}, \eqref{eq:U} and \eqref{eq:C} of Fact~\ref{fact:EquationsForZ}. We do this in order of difficulty: first the easiest Cap Equation \eqref{eq:C}, then Unitarity \eqref{eq:U} (assuming \eqref{eq:R4} and compatibility), then Reidemeister 4 \eqref{eq:R4} (hard). Finally, we prove compatibility, which is easier again, given the machinery developed for \eqref{eq:R4}.

\begin{proposition}\label{prop:CapEq} 
For any choice of $C\in \calA^{sw}(\downcap)$, the values $V_\beta$ and $C$ verify \eqref{eq:C}.
\end{proposition}
\begin{proof}
Substituting $V_\beta$ and $C$ into the Cap equation \eqref{eq:C}, we need to show that
$$u(C)(u(C))^{-1}(\varphi(p_1p_3\beta^w))^{-1}C_1C_2=C_1C_2,$$
in $\calA^{sw}(\bcap_2)$.
We cancel $u(C)(u(C))^{-1}$ on the left, and multiply on the right by $C_1^{-1}C_2^{-1}$ (this is valid as $\calA(\bcap_2)$ is a right $\calA(\uparrow_2)$-module by stacking). Then we only need to show that
$(\varphi(p_1p_3\beta^w))^{-1}=1$ in $\calA(\downcap_2)$. We can multiply on the right by $\varphi(p_1p_3\beta^w)$, hence it's enough to see that $1=\varphi(p_1p_3\beta^w)$. This, in turn, is clear by the CP relation since all heads are below all tails in the image of $\varphi$. \qed
\end{proof}
\medskip

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

\begin{definition}\label{def:norm}
For an arrow diagram $D \in \calA^{sw}(\uparrow_n)$ let $D^*$, the {\em adjoint} of $D$, be the arrow diagram $A_1A_2...A_n(D)$, where $A_i$ denotes the {\em adjoint} operation applied to strand $i$. For a group-like diagram\footnote{While the definition of ``norm'' makes sense for all arrow diagrams, it is most useful, and only used, in the context of group-like diagrams.} $D=e^d$, where $d\in \calP^{sw}(\uparrow_n)$, denote by $\| D\|$ the diagram $DD^*$, and call it the ``{\em norm}\footnote{We use the word ``norm'' only because it is notationally intuitive; we do not claim that $\| D\|$ satisfies the properties of a norm. The reader might object that it should be called ``norm squared''; in our opinion this would clutter up the notation too much.}'' of $D$.   
\end{definition}


\begin{lemma}\label{lem:Norms} The adjoint operation and norm satisfy the following basic properties:
\begin{enumerate}
\item\label{pr:mult} $(D_1D_2)^*=D_2^*D_1^*$.
\item\label{pr:exp} If $d \in \calP^{sw}(\uparrow_n)$ is a primitive arrow diagram, then $(e^d)^*=e^{d^*}$.
\item \label{pr:wheel} If $d \in \calP^{sw}(\uparrow_n)$ is a wheel, then $d^*=d$, and $\|e^d\|= e^{2d}.$
\item \label{pr:tree} If $d \in \calP^{sw}(\uparrow_n)$ is a tree, then $\|e^d\|\in \exp(\langle \text{wheels} \rangle)$.
\item\label{pr:product} If $d_1, d_2\in \calP^{sw}(\uparrow_n)$, then $\|e^{d_1}e^{d_2}\|=\|e^{d_1}\|\|e^{d_2}\|^{e^{d_1^*}}=\|e^{d_1}\|\|e^{d_2}\|^{e^{-d_1}}$, where the exponent denotes conjugation.
\item \label{pr:Ahor} If $d\in \calP^{hor}(\uparrow_n)$ is a primitive horizontal chord diagram, then $\|e^{\alpha(d)}\|=1$. 
\end{enumerate}
\end{lemma}

\begin{proof}
Properties \eqref{pr:mult}, \eqref{pr:exp}, and \eqref{pr:wheel} are immediate from the definitions. Property \eqref{pr:tree} follows from the fact that for a tree arrow diagram $d$, the sum $(d+d^*)$ is a linear combination of wheels by repeated applications of $\aSTU$ relations.

We verify property \eqref{pr:product} directly:
$$\|e^{d_1} e^{d_2}\|=e^{d_1}e^{d_2}e^{d_2^*}e^{d_1^*}=e^{d_1} e^{d_1^*}e^{-d_1^*}e^{d_2}e^{d_2^*}e^{d_1^*}=\|e^{d_1}\|\|e^{d_2}\|^{e^{d_1^*}}.$$
To show the second equality, note that by definition $e^{d_1^*}=e^{-d_1}\|e^{d_1}\|$. By Property \eqref{pr:tree}, $\|e^{d_2}\|\in \exp(\langle \text{wheels} \rangle)$, so $\|e^{d_2}\|^{e^{-d_1}}\in \exp(\langle \text{wheels} \rangle)$. Since  $\|e^{d_1}\| \in \in \exp(\langle \text{wheels} \rangle)$, it acts trivially on $\|e^{d_2}\|^{e^{-d_1}}$.

To prove property \eqref{pr:Ahor} observe that for a single chord $t_{ij}\in \calA^{hor}$, $\alpha(t_{ij})=a_{ij}+a_{ji}$, that is, the sum of arrows from strand $i$ to strand $j$ and vice versa. Therefore, $(\alpha(t_{ij}))^*=-\alpha(t_{ij})$. Furthermore, if in $\calP^{sw}(\uparrow_n)$, $x$ and $y$ are two primitive arrow diagrams such that $x^*=-x$ and $y^*=-y$, then by direct computation we also have $([x,y])^*=-[x,y]$. Since the set $\{(t_{ij})\}_{1\leq i<j \leq n}$ generate $\calP^{hor}(\uparrow_n)$ as a Lie algebra, this implies that $(\alpha(d))^*=-\alpha(d)$ for any $d\in \calP^{hor}(\uparrow_n)$. Thus, by direct computation 
$$\|e^{\alpha(d)}\|=e^{\alpha(d)}e^{(\alpha(d))^*}=e^{\alpha(d)}e^{-\alpha(d)}=1.$$
\qed
\end{proof}

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

\begin{theorem}\label{thm:Unitarity}
Suppose that $Z^{tree}: \wTFe \to \calA^{tree}$ is a filtered map with $Z^{tree}(\pvertex):=V^{tree}$, such that $V^{tree}$ satisfies the \eqref{eq:R4} equation, and for any $K\in \sKTG$, $Z^{tree}(a(K))=\pi\alpha Z^u(K)$. Then there exists a group-like $C\in \calA^{sw}(\downcap)$ which, along with $V:=C_1^{-1}C_2^{-1}V^{tree}C_{12}$, defines a homomorphic expansion $Z^w: \wTFe \to \calA^{sw}$.
\end{theorem}

\begin{proof}
Indeed, the values $(V,C)$ uniquely determine $Z$. To show that $Z$ is a homomorphic expansion, we need to prove that $(V,C)$ satisfy the \eqref{eq:R4}, \eqref{eq:C} and \eqref{eq:U} equations. The \eqref{eq:R4} equation only depends on $V^{tree}$. Since $\pi(V)= \pi(C_1^{-1}C_2^{-1}V^{tree}C_{12})=V^{tree}$, therefore $V$ satisfies \eqref{eq:R4} by assumption. The fact that $(V,C)$ satisfy the Cap \eqref{eq:C} equation follows from Proposition~\ref{prop:CapEq}.

For the remainder of this proof we denote $V^{tree}=:T$. The main difficulty is to show that $(V,C)$ satisfy the Unitarity \eqref{eq:U} equation, which can be re-stated as $\|V\|=1$. Substituting $V=C_1^{-1}C_2^{-1}V^{tree}C_{12}$, the \eqref{eq:U} equation becomes $C_1^{-1}C_2^{-1}TC_{12}^2 T^* C_2^{-1}C_1^{-1}=1$, and by rearranging, this simplifies to
\begin{equation}\label{eq:Unitarity1}
\|T\|=\left( \left((C_{12})^{T^{-1}}\right)^{-1}C_1C_2  \right)^2,
\end{equation}
where $T^{-1}$ in the exponent denotes conjugation. We need to prove that there exists a value $C=e^c$, which satisfies this equation. Note that $\|T\|$ is an exponential of wheels in $\calA^{sw}(\uparrow_2)$, so this makes semantic sense.  By Fact~\ref{fact:semidirect}, $\|T_{12}\|$ is identified with an element of $\exp(\cyc\langle x,y\rangle)$ and thus represented by an exponential in cyclic words in two variables. Let $f(x,y)\in \cyc\langle x,y \rangle$ denote the logarithm of $\|T\|$. In turn, $c \in \cyc\langle x \rangle \cong \bbQ[[x]]$, thus we write $c=c(x)$. 

We claim that $$(C_{12})^{T^{-1}}=(\exp(c(x+y)))^{T^{-1}}=T\exp(c(x+y))T^{-1}= \exp(c(\log(e^xe^y))).$$ The last equality follows from the fact that $T$ satisfies the \eqref{eq:R4} equation; this is explained in detail in \cite{Bar-NatanDancso:WKO2}, Section 4.4, in particular in and around Fig.\ 19.

With this reduction in place \eqref{eq:Unitarity1} reduces to 
$$e^{f(x,y)}=\left(e^{c(\log(e^xe^y))}e^{c(x)}e^{c(y)}\right)^2,$$
and since all exponents (wheels) commute, this is equivalent to
\begin{equation}\label{eq:Unitarity2}
f(x,y)=2\big(c(x)+c(y)+c(\log(e^xe^y))\big).
\end{equation}
which we need to solve for $c\in \bbQ[[x]]$.



\parpic[r]{\input figs/phig.pstex_t }
To do so, we use that $Z^{tree}(a(K))=\pi\alpha Z^u(K)$, specifically
for the associator graph $K=\phi$ shown on the right. Recall
that $Z^u(\phi)$ is a Drinfel'd associator $\Phi$. On the other hand,
$\alpha(\phi)$ is a circuit algebra product of four vertices, therefore
$$\alpha(\Phi)=Z^{tree}(a(\phi))=T_{12,3}^{-1}T_{12}^{-1}T_{23}T_{1,23}.$$
Rearranging, we obtain
$$T_{12}T_{12,3}\alpha(\Phi)=T_{23}T_{1,23}.$$

Applying the ``norm'' to both sides, keeping in mind that $\|\alpha(\phi)\|=1$ by Lemma~\ref{lem:Norms} property \eqref{pr:Ahor}, and using the exponent notation for the adjoint action on a norm, we get
\begin{equation}\label{eq:UnitarityMidProof}
\|T_{12}\| \|T_{12,3}\|^{T_{12}^{-1}}=\|T_{23}\| \|T_{1,23}\|^{T_{23}^{-1}}.
\end{equation}
Recall that $\|T_{12,3}\|=\exp(f(x+y,z))$, $\|T_{23}\|=\exp(f(y,z))$, $\|T_{1,23}\|=\exp(f(x,y+z))$.

As before, we have $$\|T_{12,3}\|^{T_{12}^{-1}}=(\exp(f(x+y,z)))^{T_{12}^{-1}}=T_{12}\exp(f(x+y,z))T_{12}^{-1}= \exp(f(\log(e^xe^y),z)),$$ since $T$ satisfies the \eqref{eq:R4} equation. Similarly, $$\|T_{1,23}\|^{T_{23}^{-1}}= (\exp(f(x,y+z)))^{T_{23}^{-1}}=\exp(f(x, \log(e^ye^z))).$$

Thus, Equation \eqref{eq:UnitarityMidProof} reduces to
$$e^{f(x,y)}e^{f(\log(e^xe^y),z)}=e^{e^{f(y,z)}}e^{f(x,\log(e^ye^z))},$$
and since all the exponentials (wheels) commute, this is equivalent to 
\begin{equation}
f(x,y)=f(y,z)+f\left(x,\log(e^ye^z)\right)-f\left(\log(e^xe^y),z\right)
\end{equation}
The fact that there exists a solution $c(x)$ to Equation~\ref{eq:Unitarity2} follows from this equation via a short cohomology calculation as in \cite[Proposition 27]{AlekseevEnriquezTorossian:ExplicitSolutions}, and \cite[Appendix A, Proof of Theorem 2.8]{AlekseevTorossian:KashiwaraVergne}.
\qed
\end{proof}

\medskip

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

The reader might ask, why work with a space as inconvenient
as $\sKTG$ instead of, say, a planar algebra of trivalent tangles? 
The answer is that the existence of a homomorphic expansion is a highly non-trivial property, and in particular 
ordinary trivalent tangles do not have one. Even without trivalent vertices, ordinary tangles, or u-tangles, do not have have a homomorphic expansion as a planar algebra\footnote{We only mention that the planar algebra of u-tangles does not have a homomorphic expansion $Z^t$ so as to explain why we are not using one. This said, the non-existance of $Z^t$ is easy to prove: by an explicit calculation in degree 2 one shows that there is no linear combination of chord diagrams that can serve as $Z^t(\overcrossing)$, satisfying the R3 relation.}. 
{\em Parenthesized tangles} (a.k.a. {\em q-tangles}) \cite{LeMurakami:Universal, Bar-Natan:NAT} do have homomorphic expansions, yet in fact these are almost equivalent to $\sKTG$ \cite{Thurston:KTG, Bar-NatanDancso:KTG, Dancso:KIforKTG}. 

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

\begin{itemize}
\item From here on we denote the value $V^{tree}$ arising from the ``buckle'' construction of Section~\ref{subsec:Part1TreeProof} by $V^{tree}_\beta$. Assuming the conditions of Theorem~\ref{thm:Unitarity} hold, there is a unique homomorphic expansion $Z^w_\beta$ for which $\pi(Z^w(\pvertex))=V^{tree}_\beta$.

\item In a new construction, we map classical trivalent tangles into $\sKTG$ via a ``{\em double tree}'' map, 
and use this to define a partial expansion $Z^w_\xi$ for the $a$-images of classical trivalent tangles. This in particular defines a new vertex value $Z^w_\xi(\pvertex)=V_\xi$.

\item After some technical lemmas, we show that $Z^w_\xi$ satisfies the compatibilty $Z^w(a(K))=\alpha Z^u(K)$ for all $K\in \sKTG$: see Proposition~\ref{prop:uwCompatibility}.

\item We show that restriction of $Z^w_\xi$ to $a(\uTT)$ is a planar algebra map (see Theorem~\ref{thm:PAMap}), and thus $Z^w_\xi$ satisfies the \eqref{eq:R4} equation.

\item We show that the two constructions yield the same vertex value on the tree level: $\pi(V_\xi)=V^{tree}_\beta$. This is done in Lemma~\ref{lem:TwoConstructions}. Thus, by Lemma~\ref{lem:PiVEnough}, if the two constructions produce homomorphic expansions, those expansions are the same.


\item Putting this together, $V^{tree}_\beta$ satisfies \eqref{eq:R4} and the compatibility condition of Theorem~\ref{thm:Unitarity}, and thus exists a unique homomorphic expansion $Z^w=Z^w_\xi=Z^w_\beta$, compatible with $Z^u$. We prove this in detail in Theorem~\ref{thm:ZwExpansion} below.
\end{itemize}


\subsubsection{The new construction: $Z^w_\xi$}\label{subsubsec:ZwDef}

We start by defining (classical, or {\em usual}) trivalent tangles,  denoted $\uTT$:  
\[
  \uTT:=\PA\!\left.\left.\left\langle
  \raisebox{-2mm}{\input{figs/uTTgens.pstex_t}}
  \right|
  \parbox{1.3in}{\centering \Rs, R2, R3, R4}
  \right|
   \parbox{0.5in}{\centering $S_e$, $u_e$}
  \right\rangle
\]


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




\parpic[r]{\input figs/TangleUnzip.pstex_t }
The elements of $\uTT$ are usual -- that is, classical -- trivalent tangles with ordered ends (the ordering is assumed to be counterclockwise from bottom left, unless otherwise stated), and signed vertices with a total ordering of edges at each vertex. 
There are no ``virtual crossings'' in planar algebras. The relations are the usual Reidemeister relations which make sense in this context (\Rs, R2, R3 and R4). 
The planar algebra $\uTT$ is equipped with auxiliary orientation switch and edge unzip operations. Edge unzips are defined for edges that connect a positive and a negative vertex in as shown in the figure on the right. The planar algebra $\uTT$ does not have a homomorphic expansion. 


\begin{figure}
\input{figs/dt.pstex_t}
\caption{The {\em double tree map}:  
connect the ends of $T$ by two binary trees (hence ``double tree''), as shown. Note that the tree on the left
always crosses over the tree on the right, and all edges of both trees are oriented towards $T$. }\label{fig:dt}
\end{figure}

We define a {\em double tree} map $\doubletree: \uTT \to \sKTG$, as in Figure \ref{fig:dt}.
The map $\doubletree$ depends on two choices of binary trees: in Figure \ref{fig:dt} we chose a particular example.
It is important that, regardless of the choice of trees, the ``left side'' tree crosses over the ``right side'' tree.
We will demonstrate that in fact the choice of trees becomes irrelevant after some post-compositions, see Lemma~\ref{lem:TreeChange}. 

Working towards a construction of $Z^w_\xi$, we post-compose $\doubletree$ with the following sequence of maps, which are explained in the paragraph below:
\begin{equation}\label{eq:dt}
T\in \uTT \stackrel{\doubletree}{\longrightarrow}\sKTG\stackrel{Z^u}{\longrightarrow}\calA^u(\doubletree(T))
\stackrel{\alpha}{\longrightarrow}\calA^{sw}(\doubletree(T))\stackrel{\kappa,u,p}{\longrightarrow}
\calA^{sw}(\check{T})\stackrel{\varphi}{\cong}\calA^{sw}(T).
\end{equation}
Here $T$ stands for an arbitrary tangle in $\uTT$. The double tree map $\doubletree$ maps $T$ into $\sKTG$, and by applying $Z^u$ one
obtains a value in $\calA^u$, namely a chord diagram on the skeleton of $\doubletree(T)$. We 
denote the space of chord diagrams on this skeleton by $\calA^u(\doubletree(T))$.
Now $\alpha$ maps this to arrow diagrams on the skeleton of $\doubletree(T)$, that is, to $\calA^{sw}(\doubletree(T))$.
In order to revert the skeleton back to that of $T$, we apply some operations in $\calA^{sw}$: a cap attachment $\kappa$, unzips and punctures (as shown in Figure \ref{fig:pCu} and 
explained below), resulting in a slightly modified version of the desired skeleton, denoted $\check{T}$. Finally, we use that 
$\calA^{sw}(\check{T})\cong\calA^{sw}(T)$ via the sorting isomorphism $\varphi$ of Lemma~\ref{lem:CapString}, and hence we obtain
a value in $\calA^{sw}(T)$, as needed, which, we will later see, is {\em almost} $Z^{w}(a(T))$. (Although the punctured strands connect in a single binary tree, VI relations can be used as part of the sorting isomorphism.)


\begin{figure}
\input{figs/pCu.pstex_t}
\caption{The cap attachment, unzips and punctures. While these
operations are applied in $\calA^{sw}$ -- there are arrows on these
skeleta -- for simplicity the figure only shows the effect on the
skeleton.}
\label{fig:pCu} \end{figure}

The cap attachment, unzip and puncture operations are done in the order shown in Figure~\ref{fig:pCu}. First attach a cap -- a capped strand with no arrows on it -- to the end of the right vertical strand in $\alpha(\doubletree(T))$:
this is a circuit algebra operation in $\calA^{sw}$. If $T$ has $n$ ends, perform $(n-1)$ consecutive disc unzips on the capped strand, 
as shown in Figure~\ref{fig:pCu}. Then puncture the  left hand tree, for example by puncturing the left vertical strands marked ``1, 2,...'' in
Figure~\ref{fig:pCu} (these punctures also affect the connecting diagonal
strands, as in Figure~\ref{fig:punctures}). 
Note that since the punctured tree had originally crossed over the 
capped tree, these crossings become virtual after puncturing, hence the last equality in Figure~\ref{fig:pCu}. 

Denote the composition of the maps and operations shown in Equation (\ref{eq:dt}) by $\xi$, that is, 
\begin{equation}\label{eq:xidef}
\xi:=\varphi \circ p\circ u \circ \kappa\circ \alpha\circ Z^u \circ \doubletree: \uTT \to \calA^{sw}.
\end{equation}
Then, $\xi(T)\in \calA^{sw}(T)$. We first show that $\xi(T)$ is well-defined,
that is, it doesn't depend on the choice of binary trees in $\doubletree(T)$.

\begin{lemma}\label{lem:TreeChange}
 The choice of binary trees in the double tree construction does not affect $\xi(T)$.
\end{lemma}

\parpic[r]{\input figs/IH.pstex_t }
{\em Proof.}
 Any binary tree can be changed into any other binary tree via a sequence of ``I to H'' moves, as shown on the right.
 Hence, it is enough to analyze how an I to H move on one of the trees affects the value of $Z^u(\doubletree(T))$,
 and prove that the difference vanishes after the caping, unzip, and puncture operations. 
 
 \parpic[r]{\input figs/IHPhi.pstex_t }
 Suppose $\tau_1$ and $\tau_2$ are two binary trees which differ by a single I to H move, and let $\doubletree_{\tau_1}$ 
 and $\doubletree_{\tau_2}$ denote the two resulting double-tree maps,
 assuming the ``other side tree'' is unchanged. The 
 I to H move can be realised by inserting\footnote{See \cite[Section 4.6]{Bar-NatanDancso:WKO2} for a detailed description of the tangle insertion operation in $\sKTG$.} an associator, followed by unzipping the edge marked `1' on the right,
 then the edge marked `2'. By the homomorphicity of $Z^u$, 
 the values $Z^u(\doubletree_{\tau_2}(T))$ and $Z^u(\doubletree_{\tau_1}(T))$, only differ in an 
 inserted horizontal chord associator $\Phi$ on the three strands involved, we indicate this by writing $Z^u(\doubletree_{\tau_2}(T))=Z^u(\doubletree_{\tau_1}(T))*\Phi$.
 If the I to H move was done on the left side tree, then all the strands involved are later punctured, killing any arrow diagram that
 lived on them by the TF relation. As a result, the only surviving part of $\Phi$ is its constant term, 1, and the resulting values of 
 $\xi$ are equal.
 

 If the I to H move is done on the right side tree, then the all participating strands 
 are capped and disk unzipped. If $\alpha(\Phi)$ is immediately adjacent to the caps, then it cancels by the $CP$ relation.
 However, it is a priori possible that there are other arrow ending separating $\Phi$ from the caps. Note that in $\calA^{u}$, any chord endings can be can be commuted from below the associator to above, using $VI$ relations and the {\em invariance property} of chord diagrams shown in Figure~\ref{fig:Invariance} \cite[Lemma 3.4]{Bar-Natan:NAT}.
 Thus, one can assume that $\alpha(\Phi)$ is adjacent to the caps and hence cancels.
 This concludes the proof. \qed
 
 \begin{figure}
 \input figs/Invariance.pstex_t
 \caption{The invariance property of chord diagrams on the left, the head invariance and tail invariance properties of arrow diagrams in the middle and right. Here $D$ denotes a chord or arrow diagram on any skeleton; for tail invariance there is a restriction that $D \in \im \alpha$ (with $\alpha: \calA^u \to \calA^{sw} $). The box labelled ``$+$'' denotes a sum of incoming chords or arrows, whose other ends are in the same place. The equalities are understood locally: there may be other skeleton components and chords/arrows elsewhere, which coincide on both sides.}\label{fig:Invariance}
 \end{figure}
 
There is an action of $\bbZ/n\bbZ$ on elements of $\uTT$ with $n$ ends, by cyclic permutations of the ends.
The following lemma will be useful later in proving that $Z^w$ is a planar algebra map; we present it now because its proof 
is similar to that of Lemma~\ref{lem:TreeChange}.

\begin{lemma}\label{lem:CyclicPerm}
 The map $\xi$ is invariant under cyclic permutation of the ends of $T$.
\end{lemma}

\begin{proof}
To show that $\xi(T)$ is invariat under cyclic permutations of ends of $T$, 
it is enough to show that $\xi(T)$ does not change when the rightmost end of $T$ is moved to the far left (denote this by $\sigma T$),
as shown in Figure \ref{fig:welldef}. 

\begin{figure}
 \input figs/WellDef.pstex_t
 \caption{Double tree construction for cyclically permuted ends of $T$.}\label{fig:welldef}
\end{figure}

The rightmost picture of Figure \ref{fig:welldef} is equivalent as $\sKTG$s to $\doubletree(\sigma T)$. It differs from 
$\doubletree(T)$ in three ways: 
\begin{itemize}
\item the binary trees connecting the ends of $T$ are different; 
\item two tree branches (marked with $*$ in Figure \ref{fig:welldef})
are connected to the trunk from the opposite side: that is, these trivalent vertices have opposite cyclic orientation; 
\item one tree branch has a kink in it.
\end{itemize}

As before, we need to analyse how $Z^u(\doubletree(\sigma T))$ differs from $Z^u(\doubletree(T))$, and show that the difference
vanishes after the puncture, cap and unzip operations.

\begin{figure}
\input{figs/WellDef2.pstex_t}
\caption{The difference between $\doubletree(T)$ and $\doubletree(\sigma T)$, understood via insertions.}\label{fig:welldef2}
\end{figure}

To achieve this, we transform $\doubletree(\sigma T)$ into $\doubletree(T)$ using tangle insertions. 
First, cancel the kink by inserting an opposite kink $I_{1}$ on the same strand,
as shown in Figure \ref{fig:welldef2} in blue\footnote{Or grey in black and white print.}.
As $Z^u$ is compatible with insertion, the $Z^u$
values will differ by the value of a kink: a chord diagram on the one strand involved. 
Later in the process this strand is 
punctured, so the value of the kink cancels by the TF relation.

Similarly, switching the side that the tree branches are attached on amounts to inserting twists $I_{2}$ and $I_{3}$, and unzipping the connecting edges, 
also shown in Figure \ref{fig:welldef2}. Each of these operations changes
the value of $Z^u$ by inserting the value of a twist, which is $e^{c/2}$ for any $Z^u$, where $c$ denotes a single chord between the appropriate strands \cite[Lemma 4.14]{Bar-NatanDancso:WKO2}. 
Applying $\alpha$ maps this to $e^{(a_L+a_R)/2}$,
where $a_L$ and $a_R$ denote horizontal left and right arrows, respectively. On the left side tree, this cancels after punctures, 
as before. 
On the right side tree, the strand directly underneath the twist is capped and uzipped, and hence the value of the twist cancels by the CP relation.

Now observe that the right side picture of Figure \ref{fig:welldef2} only differs from $\doubletree(T)$ in the choices of binary trees,
which do not change the value of $\xi$ by Lemma \ref{lem:TreeChange}.  \qed
\end{proof}

\vspace{2mm}
The following lemma clarifies the relationship between the map $\xi$ and the (partial) homomorphic expansion $Z^w_\xi$ that we're aiming to construct:

\begin{lemma}\label{lem:Compatibility}
 If there exists a homomorphic expansion $Z^w_\xi$ for $\wTFe$ compatible with $Z^u$,
 and $T\in \uTT$ is a tangle with $n$ ends, then $Z^w_\xi(a(T))=\xi(T)\cdot (C^{-1})^n$, where $C=Z(\hspace{-0.7mm}\bcap\hspace{0.5mm})$, and $\xi(T)$ is multiplied by $C^{-1}$ at each 
 tangle end of $T$, as in Figure~\ref{fig:ZTCheck}.
\end{lemma}

\begin{proof}
Assume there exists a homomorphic expansion $Z^w_\xi$ compatible with $Z^u$. We use, as in Figure~\ref{fig:BigCompat},
the homomorphicity of $Z^w_\xi$ and its compatibility with $Z^u$ to show that
$\xi(T)=Z^w_\xi(\check{T})$, where $\check{T}$ is as in Equation~\ref{eq:dt} and shown in Figure~\ref{fig:ZTCheck} on the left.




\begin{figure}
$ \xymatrix{
T \ar@{}[d]|-*[@]{\text{\begin{rotate}{-90}$\in$\end{rotate}}} \ar@{}[rr]|-*[@]{\mapsto} 
&
& \doubletree(T) \ar@{}[d]|-*[@]{\text{\begin{rotate}{-90}$\in$\end{rotate}}}  \ar@{}[r]|-*[@]{\mapsto} 
& \doubletree(T) \ar@{}[d]|-*[@]{\text{\begin{rotate}{-90}$\in$\end{rotate}}}  \ar@{}[r]|-*[@]{\mapsto} 
& \kappa \doubletree(T) \ar@{}[d]|-*[@]{\text{\begin{rotate}{-90}$\in$\end{rotate}}}  \ar@{}[r]|-*[@]{\mapsto} 
& \hat{T} \ar@{}[d]|-*[@]{\text{\begin{rotate}{-90}$\in$\end{rotate}}} 
\\ 
\uTT  \ar[rr]^{\doubletree} \ar `d[ddr][ddrrrrr]^\xi
&
&  \sKTG \ar[r]^{a}  \ar[d]^{Z^{u}}  \ar@{}[dr] |{\color{blue}{(1)}}
& \wTFe  \ar[r]^{\kappa}  \ar[d]^{Z^{w}}  \ar@{}[dr] |{\color{blue}{(2)}}
& \wTFe \ar[r]^{p\circ u}  \ar[d]^{Z^{w}} \ar@{}[dr] |{\color{blue}{(3)}}
& \wTFe  \ar[d]^{Z^{w}}
\\
&
& \calA^{u}(\doubletree(T)) \ar[r]^{\alpha}
& \calA^{sw} (\doubletree(T)) \ar[r]^{\kappa\circ \hat{C}}
& \calA^{sw}(\kappa\doubletree(T)) \ar[r]^{p \circ u}
& \calA^{sw}(\hat{T})  \ar[d]^{\varphi}
\\ 
\ar@{}[urr]|{\color{blue}{(4)}}
&&&&&\calA^{sw}(T)
}$ 
 \caption{Comparing $\xi$ and $Z^w_\xi$, assuming that $Z^w_\xi$ exists.}\label{fig:BigCompat}
\end{figure} 


If the diagram in Figure~\ref{fig:BigCompat} commutes, then for any $T\in \uTT$ and any $Z^u$-compatible $Z^w_\xi$, we have $\varphi(Z^w_\xi(\check{T}))= \xi(T)$. 
Since $Z^w_\xi$ is a circuit algebra homomorphism, $Z^w_\xi(\check{T})$ can be obtained from $Z^w_\xi(T)$ 
by attaching the $Z^w_\xi$-value of a left-punctured right-capped vertex at each tangle end, as illustrated in Figure~\ref{fig:ZTCheck}.
By Lemma \ref{lem:pV} we have $Z^w_\xi\Big(\raisebox{-2mm}{\input{figs/PunctVertex.pstex_t}}\Big)=1$, so the only additions are $C$ values at each capped end, 
as shown in Figure~\ref{fig:ZTCheck}. This can then be interpreted as a value in $\calA^{sw}(T)$ via the sorting isomorphism $\varphi$ of Lemma~\ref{lem:CapString}. This implies the statement of the Lemma.


\begin{figure} 
 \input figs/ZTCheck.pstex_t
 \caption{Computing $Z^w_\xi(\check{T})$ and $Z^w_\xi(T)$.}
 \label{fig:ZTCheck}
\end{figure}


It remains to show that the diagram in Figure~\ref{fig:BigCompat} commutes. The square $(1)$ is the assumed the compatibility of $Z^u$ and $Z^w_\xi$. 
In square $(2)$, recall the map $\kappa$ denotes the circuit algebra operation of 
attaching a cap at the bottom right end of the w-foam. The map $\hat{C}$ denotes the circuit algebra operation which attaches a 
value $C=Z(\hspace{-0.7mm}\bcap\hspace{0.5mm})$ at the end of the strand. Thus, the commutativity of square $(2)$ is implied by the homomorphicity of $Z^w_\xi$ with respect to circuit algebra composition (as a binary operation). 
The square $(3)$ is commutative due to the homomorphicity of $Z^w_\xi$ with respect to punctures and disc unzips.


\parpic[r]{\input figs/MoveCap.pstex_t }
The commutativity of the heptagon $(4)$ would be true by definition, if not for the map $\hat{C}$ (multiplication by the cap value). We show that, in fact, the value $C$ cancels after punctures, by
a property of arrow diagrams in the image of $\alpha$, called {\em tail-invariance}, shown in Figure~\ref{fig:Invariance} (see \cite{Bar-NatanDancso:WKO2}, Remark 3.14 and early in Section 3.3). In the current situation tail invariance means
that the value $C$, which has only arrow tails, can be moved from one tangle end to the other, as shown on the right. Consequently, $C$ cancels when
the left strand is punctured. 
 \qed
\end{proof} 

\begin{remark}\label{rem:CapOrientations}
In Lemma~\ref{lem:Compatibility} we assume by convention that all tangle ends of $T$ are oriented upwards (towards $T$). If $k$ tangle ends are oriented down, the corresponding cap values appear with their orientations switched: $Z^w_\xi(aT)=\xi(T)\cdot (C^{-1})^{n-k}(S(C)^{-1})^k$.
\end{remark}

\begin{corollary}\label{cor:ZTreefromxi}
If there exists a homomorphic expansion $Z^w$ for $\wTFe$ compatible with $Z^u$, then $\pi(V)=\pi(\xi(\FlippedYGraph))$, where $V$ is the $Z^w$-value of the vertex, and $\pi$ is the tree projection. \end{corollary}

\vspace{2 mm}
Next, we aim to show that $Z^w_\xi$ is compatible with $Z^u$. This requires a technical lemma, in which we compute the $\xi$-value of a vertical strand:


\begin{lemma}\label{lem:xiofstrand}
 For a single un-knotted strand, $\xi(\curvearrowright)=\alpha(\nu^{1/2})$, where $\nu \in \calA^u(\curvearrowright)$ 
 denotes the Kontsevich integral of the un-knot\footnote{The value of $\nu$ was conjectured in
 \cite{Bar-NatanGaroufalidisRozanskyThurston:WheelsWheeling} and proven in \cite{Bar-NatanLeThurston:TwoApplications}. Note that $\nu$ involves wheels only.}.
\end{lemma}

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

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


\begin{figure}
 \input figs/dtstrand.pstex_t
 \caption{The double tree map composed with $Z^u$, applied to a single strand. To compute $Z^u(\doubletree(\curvearrowright))$, we write $\doubletree(\curvearrowright)$ as a composition of generators; this requires first expressing it as a bottom-top tangle. See \cite[Proposition 4.13]{Bar-NatanDancso:WKO2} for details.}
 \label{fig:dtstrand}
\end{figure}

In $\xi(\curvearrowright)$, $Z^u$ is followed by $\alpha$, a cap attachment, unzips and punctures.
As explained in 
\cite[Section 4.6]{Bar-NatanDancso:WKO2}, we have $\alpha(b)=e^{a/2}\alpha(\nu)^{1/2}$, and $\alpha(n)=e^{-a/2}\alpha(\nu)^{1/2}$. Both factors are local arrow diagrams on a single strand and hence commute with all other arrow endings. Hence the exponential part of $n$
cancels by the CP relation, and the wheels part $\alpha(\nu)^{1/2}$ can be moved to the bottom left end by the tail invariance property
(as in the last paragraph of the proof of Lemma~\ref{lem:Compatibility}), where it cancels given punctures.  Subsequently, the associator $\alpha(S_3(\Phi^{-1}))$ cancels by Fact~\eqref{it:Concat} of Lemma~\ref{lem:PhiFacts}. 





\begin{figure}
 \input figs/dtstrand2.pstex_t
 \caption{The value of $pu\kappa \alpha Z^u\doubletree(\curvearrowright)$: the numbering (1) through (6) refer to the paragraphs where each component is explained.}
 \label{fig:dtstrand2}
\end{figure}

Taking these cancellations into account, the value $pu\kappa \alpha Z^u\doubletree(\curvearrowright)\in \calA^{sw}$ is shown in Figure \ref{fig:dtstrand2} and explained below. 
Recall that $\alpha$ maps a chord to the sum of its two possible orientations. However, when one supporting strand is punctured, 
only one of these orientations survive. 
Hence, for example, $p_2(\alpha(R_{23}))=(e^{a_{32}/2})$. 
Figure~\ref{fig:dtstrand2} shows a schematic picture of $pu\kappa\alpha Z^u\doubletree(\curvearrowright)$ with exponentials and associators indicated 
by single arrows. Recall that 
$\Phi \in \calA^{hor}(\uparrow_3)$ can be written as a power series in any two of the three generators of $\calA^{hor}(\uparrow_3)$: $c_{12}$,
$c_{23}$ and $c_{13}$. For example, $\Phi(c_{12},c_{23})=\Phi(c_{12}, -c_{12}-c_{23})$. For each associator we chose the presentation in which $pu\kappa\alpha(\Phi)$ is of the simplest form. 

\begin{enumerate}
\item The top associator of Figure~\ref{fig:dtstrand}, after applying a VI relation, is written as
 $\Phi^{-1}_{13(24)}$ in the strand numbering of Figure~\ref{fig:dtstrand2}. We write this
in terms of $c_{13}$ and $c_{1(24)}=c_{12}+c_{14}$. Observe that $p_1\alpha(c_{13})=a_{31}$ and $p_1p_2\alpha(c_{1(24)})=a_{41}$, thus $$p_1p_2\alpha \Phi^{-1}(c_{13}, -c_{13}-c_{1(24)})=\Phi^{-1}(a_{31},-a_{31}-a_{41}).$$
This is reflected in Figure \ref{fig:dtstrand2} in drawing only the $a_{31}$ and $a_{41}$ arrows.  
Now note that by the VI relation at the top right vertex, $a_{41}=-a_{51}$. The local arrow diagrams $e^{a/2}$ and $\alpha(\nu^{1/2})$ on strand 5 commute with arrow tails, and therefore $a_{51}$ can be pulled to the left, where by another VI relation, $a_{41}=-a_{51}=-a_{31}$. Thus, $\Phi^{-1}(a_{31},-a_{31}-a_{41})= \Phi^{-1}(a_{31},0)=1$. Note the similarity to Fact~\eqref{it:Concat} of Lemma~\ref{lem:PhiFacts}.



\item Second from top, by the same argument, we have $$p_1p_2\alpha(\Phi_{324})=p_1p_2\alpha(\Phi(c_{23},c_{24}))=\Phi(a_{32},a_{42})=\Phi(a_{32},-a_{32})=1.$$


\item For the exponential, $$p_1p_2\alpha(R_{23})=p_1p_2\alpha(e^{c_{23/2}})=e^{a_{32}/2}.$$

\item Using the same reasoning again,
$$\quad \quad \quad \, p_1p_2\alpha(\Phi^{-1}_{234})=p_1p_2\alpha(\Phi^{-1}(c_{23},-c_{23}-c_{24}))=\Phi^{-1}(a_{32},-a_{32}-a_{42})=\Phi^{-1}(a_{32},0)=1.$$
Note that here we use that the arrow tail of $a_{32}$ commutes with the arrow tails of the exponential.

\item By Fact~\eqref{it:TwoPunctures} of Lemma~\ref{lem:PhiFacts},
$$p_1p_2\alpha(\Phi)_{12(34)}=1.$$ 

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

\end{enumerate}
In summary, $\xi(\curvearrowright)=\alpha(\nu^{1/2})$, as claimed.
\qed

\medskip
As an aside, Lemma~\ref{lem:xiofstrand} enables a quick computation of the even part of $C=e^c=Z^w_\xi(\downcap)$.
Recall that $c$ is a linear combination of wheels: $c=\sum_{n=2}^\infty \gamma_n w_n$. Let $c=c_0+c_1$, where $c_0$ denotes the even part of $c$ (sum of all even wheels), and $c_1$ denotes the odd part, that is, $c=c_0+c_1$. Let $C_0=e^{c_0}$, the even part of $C$. Corollary~\ref{cor:CapValue} below shows in particular that the even part of the $C$ is independent of the choice of $Z^b$ (that is, the choice of Drinfel'd associator) and $Z^u$. 

\begin{corollary}\label{cor:CapValue}
If $C=Z^w_\xi(\downcap)$, and $C_0$ is the even part of $C$, then $C_0=\alpha(\nu^{1/4})$, regardless of the choice of expansion $Z^u$ used to construct $Z^w_\xi$.
\end{corollary}

%\parpic[r]{\input figs/OneStrand.pstex_t }

\begin{proof}
By Lemma~\ref{lem:Compatibility} and Remark~\ref{rem:CapOrientations}, we have
$Z^w_\xi(\curvearrowright)= C^{-1} \xi(\curvearrowright) S(C^{-1})$. Note that $S(w_{2k})=w_{2k}$ and $S(w_{2k+1})=-w_{2k+1}$, and hence $S(C)=e^{c_0-c_1}$. Also, by homomorphicity, $Z^w_\xi(\curvearrowright)=1$. Thus, by Lemma~\ref{lem:xiofstrand},
$1= e^{c_0+c_1}\alpha(\nu^{1/2})e^{c_0-c_1}$, and therefore $\alpha(\nu^{1/2})=e^{2c_0}$, which gives $C_0=e^{c_0}=\alpha(\nu^{1/4})$. 
\end{proof}


\medskip

Next we prove that $Z^w_\xi$ is indeed compatible with $Z^u$:

\begin{proposition}\label{prop:uwCompatibility}
For any $K\in \sKTG$,  $Z^w_\xi(a(K)) = \alpha(Z^u(K))$. 
\end{proposition}

\begin{proof}
Note that $\sKTG \subseteq \uTT$, and for $K\in \sKTG$, $\doubletree(K)$ can be obtained by inserting $K$ into the top strand of $\doubletree(\curvearrowright)$: see Figure~\ref{fig:InsertK}. Since $Z^u$ is compatible with insertions, $Z^u(\doubletree(K))$ can be obtained by $Z^u(K)$ inserted into $Z^u(\doubletree(\curvearrowright))$. Through the sequence of $\alpha$, capping, puncturing, $\varphi$ and multiplications by $C^{-1}$, all of $Z^u(\doubletree(\curvearrowright))$ cancels, as in Lemma~\ref{lem:xiofstrand}. Note that the cancellations still go through despite the fact that $\alpha(Z^u(K))$ is inserted on the top strand: this follows from the fact that $\alpha(Z^u(K)) $ is in the $\alpha$-image of $\calA^u$, and the appropriate ``commutativity'' property holds in $\calA^u$. Hence, $Z^w_\xi(K)=Z^w_\xi(K)$ as required.
\qed
\end{proof}

\begin{figure}
\input figs/InsertK.pstex_t
\caption{For $K\in \sKTG$, $\doubletree(K)$ is $K$ inserted into $\doubletree(\curvearrowright)$.}\label{fig:InsertK}
\end{figure}
 
\medskip 
Now,
we show that $Z^w_\xi$ it is a planar algebra homomorphism on $a(\uTT)$. This is technically challenging, but it implies the R4 equation immediately. For the proof, it will be necessary to know the behaviour of $Z^u$ with respect to edge deletions. When an edge $e$ of a knotted trivalent graph $K\in \sKTG$ is deleted, the two vertices  at each end of $e$ cease to be vertices. The associated graded operation on chord diagrams deletes skeleton edge $e$, and chord diagrams with any chord endings on $e$ are set equal to 0.

\begin{fact}\label{fact:ZuDelete}
$Z^u$ commutes with edge deletions up to a possible correction term of $e^{\pm c/4}\nu^{1/2}$ depending on the position of the edge, as in Figure~\ref{fig:ZuDelete}.
\end{fact}

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

\noindent
{\em Proof sketch.}
In \cite[Section 4.6.1]{Bar-NatanDancso:WKO2}  $Z^{u}$ is constructed from an invariant $Z^{old}$ by adding vertex normalizations, as shown in Figure~\ref{fig:ZuDelete}. Note that the top two diagrams in the figure differ from \cite[Figure 29]{Bar-NatanDancso:WKO2} in a single edge orientation switch, which switches the vertex sign and accordingly the normalization\footnote{The point of the normalization is to make $Z^u$ commute with unzips. The reader might wonder, why normalize so that the expansion respects unzips, rather than deletions? The answer is that for finite generation of knotted trivalent graphs, unzips are crucial but deletions are not.}. In fact, $Z^{old}$ commutes with edge deletions
\cite[Proposition 6.7]{Dancso:KIforKTG}, so the edge deletion error (and hence, the correction term) for $Z^{u}$ arises from the vertex normalisations implemented, as shown in Figure~\ref{fig:ZuDelete}. 
\qed
\medskip

\begin{remark}\label{rmk:wDelete}
There is also an ``edge delete'' operation of $\wTFe$: this is not required for the finite presentation of $\wTFe$ or $\calA^{sw}$, but it is necessary for the proof of Theorem~\ref{thm:PAMap} below. When deleting an edge in $\wTFe$ -- which can be either a tube or a string --
the vertices at either end\footnote{It is also possible to delete a capped edge.} cease being vertices.  The associated graded operation $d_e: \calA^{sw}\to \calA^{sw}$ deletes the skeleton edge $e$ and sends any arrow 
diagram with arrow endings on the deleted strand to zero. The crucial fact we need is that edge delete operations for chord and arrow diagrams are compatible via the map $\alpha$, which is immediate from the definitions:
\[
\xymatrixcolsep{4pc}\xymatrixrowsep{4pc}\xymatrix{
\calA^{u} \ar[r]^{\alpha}  \ar[d]^{d_e} 
& \calA^{sw} \ar[d]^{d_e}
\\
\calA^{u} \ar[r]^{\alpha}  
& \calA^{sw} 
}
\]
\end{remark}

\begin{theorem}\label{thm:PAMap}
 The map $Z^w_\xi$ is a planar algebra map on $a(\uTT)$.
\end{theorem}

\begin{proof}
Planar algebra operations can be written as compositions of two simpler, basic operations: disjoint unions and contractions.
In the disjoint union of two tangles $T_1$ and $T_2$, the ends of $T_1 \sqcup T_2$ 
are ordered by declaring that the ordered ends of $T_1$ come first, followed by the ordered ends of $T_2$. The contraction operation
$c_i$ applies to any tangle with at least $i+1$ ends: it acts by joining the $i$-th and $(i+1)$-st ends of $T$ and 
re-numbering the rest, resulting in a tangle with two ends less. Both operations are shown in Figure \ref{fig:AtomicPAOps}. 

\begin{figure}
 \input figs/AtomicPAOps.pstex_t
 \caption{Basic planar algebra operations: disjoint union and contraction.}
 \label{fig:AtomicPAOps}
\end{figure}

Thus, we only need to show that $Z^w_\xi$ commutes with these two operations, that is, \linebreak $Z^w_\xi(T_1 \sqcup T_2)=Z^w_\xi(T_1)\sqcup Z^w_\xi(T_2)$,
and $Z^w_\xi(c_i(T))=c_i(Z^w_\xi(T))$. Note that the right sides of these equalities make sense: arrow diagrams on the skeleta of $a(\uTT)$, where $Z^w_\xi$ takes values, also form a planar algebra, and in particular disjoint union and concatenation of arrow diagrams is well defined. 

{\bf Disjoint unions.} We need to compute $\xi(T_1\sqcup T_2)$, where $T_1,T_2\in \uTT$. 
The value $\doubletree(T_1\sqcup T_2)$  is shown in Figure~\ref{fig:DisjUni}. The binary trees in $\doubletree$ can be chosen 
arbitrarily by Lemma~\ref{lem:TreeChange}: Figure~\ref{fig:DisjUni} shows the most convenient trees for this proof. 

Observe that $\doubletree(T_1\sqcup T_2)$ can be obtained as an
$\sKTG$ by inserting $\doubletree(T_1)$ and $\doubletree(T_2)$ into a simpler $\sKTG$, denoted $H$, as shown in the same figure (up to 
orientation switches which don't impact what follows and will be ignored). Hence, $Z^u(\doubletree(T_1\sqcup T_2))$
is given by inserting $Z^u(\doubletree(T_1))$ and $Z^u(\doubletree(T_2))$ into $Z^u(H)$. 

One could compute $Z^u(H)$ explicitly using the same algorithm 
as before, but we can avoid this work, as follows. All chords in $Z^u(H)$ can be assumed to be located in the rectangle shown in 
Figure~\ref{fig:DisjUni} (using VI relations, if necessary). During the computation of $\xi$ both supporting strands are punctured, and therefore
$p^2\alpha(Z^u(H))=1$. This implies that $\xi(T_1\sqcup T_2)=\xi(T_1) \sqcup \xi(T_2)$, and it follows via Lemma~\ref{lem:Compatibility} that $Z^w_\xi(T_1\sqcup T_2)=Z^w_\xi(T_1) \sqcup Z^w_\xi(T_2)$.

\begin{figure}
 \input figs/DisjUni.pstex_t
 \caption{The double tree map applied to a disjoint union of $\uTT$-s is the same as inserting the double tree of each individual $\uTT$ into
 the $\sKTG$ $H$. In $Z^u(H)$ all chords can be pushed into the rectangle shown, using VI relations when necessary.}
 \label{fig:DisjUni}
\end{figure}


\begin{figure}
$ \xymatrixcolsep{3.5pc}\xymatrixrowsep{4pc}\xymatrix{
\uTT  \ar[r]^{\doubletree} \ar[d]^{c}   \ar@{}[dr] |{\color{blue}{(1)}}
&  \sKTG \ar[r]^{Z^u}  \ar[d]^{d^3}   \ar@{}[dr] |{\color{blue}{(2)}}
& \calA^{u}  \ar[r]^{\alpha}  \ar[d]^<<<<{(\cdot e^{-\frac{c}{4}}\nu^{\frac{1}{2}})\circ d^3}  \ar@{}[dr] |{\color{blue}{(3)}}
& \calA^{sw} \ar[r]^{\kappa}  \ar[d]^<<<<{(\cdot e^{-\frac{a}{2}}\alpha(\nu^{\frac{1}{2}}))\circ d^3}  \ar@{}[dr] |{\color{blue}{(4)}}
& \calA^{sw} \ar[r]^{u^{n-1}} \ar[d]^<<<<{(\cdot e^{-\frac{a}{2}}\alpha(\nu^{\frac{1}{2}}))\circ d^3}  \ar@{}[dr] |{\color{blue}{(5)}}
& \calA^{sw} \ar[r]^{p^{n-2}}  \ar[d]^<<<<{(\cdot e^{-\frac{a}{2}}\alpha(\nu^{\frac{1}{2}}))\circ d^3}  \ar@{}[dr] |{\color{blue}{(6)}}
& 
\\
\uTT  \ar[r]^{\doubletree}  %\ar@/_4.0pc/[ddrrrrr]^{\xi} 
&  \sKTG \ar[r]^{Z^u}  
& \calA^{u}  \ar[r]^{\alpha} 
& \calA^{sw} \ar[r]^{\kappa}  
& \calA^{sw} \ar[r]^{u^{n-3}} 
& \calA^{sw} \ar[r]^{p^{n-2}}  
& 
}$ 

\vspace{6mm}
\hspace{-11.4cm}
{\em diagram continues...}

\vspace{-7.5mm}
\hspace{4.3cm}
$
\xymatrixcolsep{4pc}\xymatrixrowsep{4pc}\xymatrix{
\ar[r]
&\calA^{sw} \ar[r]^{p^{2}}  \ar[d]^<<<<{(\cdot e^{-\frac{a}{2}}\alpha(\nu^{\frac{1}{2}}))\circ d^3} \ar@{}[drr] |{\color{blue}{(7)}}
& \calA^{sw} \ar[r]^{\varphi}  
& \calA^{sw} \ar[r]^{\cdot C^{-(n-1)}SC^{-1}}  \ar[d]^<<<<{(\cdot \alpha(\nu^{-\frac{1}{2}}))\circ c}  \ar@{}[dr] |{\color{blue}{(8)}}
& \calA^{sw} \ar[d]^{c}
\\
\ar[r]
&\calA^{sw} \ar[rr]^{\varphi}  
& 
& \calA^{sw} \ar[r]^{\cdot C^{-(n-2})}  
& \calA^{sw} 
}
$

%\vspace{1cm}\hspace{-5mm}
 %\input figs/ContractCommutes.pstex_t
 \caption{Summary of the proof that $Z^w_\xi$ commutes with contractions: $Z^w_\xi$ is the composition along the entire top and entire bottom horizontal edge of the diagram.}
 \label{fig:ContractCommutes}
\end{figure} 
 


{\bf Contractions.} Proving that $Z^w_\xi$ commutes with contractions is more involved. By Lemma~\ref{lem:CyclicPerm}, we 
can assume that the ends contracted are the last (rightmost) two ends of the $n$ ends of $T$. Hence we will drop the subscript from
$c_i$ and denote this operation simply by $c$. 

We need to show that $Z^w_\xi(cT)=cZ^w_\xi(T)$, for any $T\in \uTT$. Since $Z^w_\xi$ is given by the composition of many maps, this can be restated as the commutativity of the perimeter of a large diagram (shown in Figure~\ref{fig:ContractCommutes}), which in turn can be broken down to its smaller parts. Throughout this proof, let $T \in \uTT$ denote an arbitrary trivalent tangle.

%\newpage
%\parpic[r]{\input figs/ContractCommute1.pstex_t }
%{\em Square (1).} This square plays out in $\uTT$ and $\sKTG$, and
%commutes by inspection, as shown on the right. The three strands to be
%deleted are indicated by broken lines. Therefore, $d^3 \doubletree (T)=
%\doubletree c(T).$
%
%{\em Square (2).} Square (2) is shown schematically below on the left: for
%the $Z^u$-values skeleta are indicated but chords are not shown. To prove
%that square (2) commutes, we use the properties of $Z^u$ with respect
%to deleting edges in $\sKTG$, as stated in Fact~\ref{fact:ZuDelete}
%and Figure~\ref{fig:ZuDelete}.
%
%\parpic[l]{\input figs/ContractCommute2.pstex_t }
%Only one of the three edge deletions requires a correction term: this
%is the edge marked with $*$ in the diagram on the left. This edge
%ends in a $e^{-c/4}\nu^{1/2}$ inserted at the place of the vertex,
%where $c$ stands for a single chord. In square (2), this correction
%term appears at the bottom right corner of the square, where the
%two ends of $T$ are contracted (see in the diagram showing skeleta
%in Figure~\ref{fig:ContractCommutes}). In summary: $ \left(d^3 Z^u
%\doubletree (T)\right) \cdot (e^{-\frac{c}{4}}\nu^{\frac{1}{2}})= Z^u
%\doubletree c(T).$

{
\def\xadt{$\xrightarrow{\doubletree}$}
\def\xaZu{$\xrightarrow{Z^u}$}
\def\dote{$\cdot e^{-\frac{c}{4}} \nu^{\frac{1}{2}}$}
\parpic[r]{\input figs/ContractCommute12.pstex_t }
{\em Square (1).} This square (shown schematically on the right)
plays out in $\uTT$ and $\sKTG$, and commutes by inspection, as shown
on the right. The three strands to be deleted are indicated by broken
lines. Therefore, $d^3 \doubletree (T)= \doubletree c(T).$

{\em Square (2).} Square (2) is also on the right: for the $Z^u$-values
skeleta are indicated but chords are not shown. To prove that
square (2) commutes, we use the properties of $Z^u$ with respect to
deleting edges in $\sKTG$, as stated in Fact~\ref{fact:ZuDelete} and
Figure~\ref{fig:ZuDelete}.

Only one of the three edge deletions requires a correction term: this
is the edge marked with $*$ in the diagram on the left. This edge
ends in a $e^{-c/4}\nu^{1/2}$ inserted at the place of the vertex,
where $c$ stands for a single chord. In square (2), this correction
term appears at the bottom right corner of the square, where the
two ends of $T$ are contracted (see in the diagram showing skeleta
in Figure~\ref{fig:ContractCommutes}). In summary: $ \left(d^3 Z^u
\doubletree (T)\right) \cdot (e^{-\frac{c}{4}}\nu^{\frac{1}{2}})= Z^u
\doubletree c(T).$

\parpic[r]{\input figs/ContractCommute3.pstex_t }
{\em Square (3).}  Square $(3)$ is essentially the commutativity of edge
deletions stated in Remark~\ref{rmk:wDelete}, combined with applying
$\alpha$ to the correction term. So we have
$ \left(d^3 \alpha Z^u \doubletree (T)\right)\cdot (e^{-\frac{a}{2}} \alpha(\nu^{\frac{1}{2}})) = \alpha Z^u \doubletree c(T).$

\begin{figure}
\input figs/ContractCommute6.pstex_t
\caption{The squares (4) (5) and (6). Strands to be deleted are drawn in dashed lines throughout. The $*$ denotes a cap of interest: see the proof paragraph on square (5).}\label{fig:456}
\end{figure}

{\em Square (4).} Squares (4), (5), and (6) are shown in detail in Figure~\ref{fig:456}. Square (4) plays out in $\calA^{sw}$ and it is commutative as the deletions and the cap attachments (denoted by $\kappa$) affect different strands: see the diagram on the right. Therefore, 
$ \left(d^3 \kappa \alpha Z^u \doubletree (T)\right)  \cdot (e^{-\frac{a}{2}}\alpha(\nu^{\frac{1}{2}}))=\kappa \alpha Z^u \doubletree c(T).$

{\em Square (5).} The only difference between $d^3 \circ u^{n-1}$ and $u^{n-3}\circ d^3$ is what happens to arrows on the caped skeleton edge marked by $*$ in Figure~\ref{fig:456}. Following the diagram right and down, this edge is unzipped $n-1$ times, then the last two of its daughter edges are deleted. On the other hand, following the diagram down and right, the same edge is unzipped $n-3$ times. The results of these compositions are the same by definition of the unzip and delete operations. Thus, we have
$$ \left(d^3 u^{n-1} \kappa \alpha Z^u \doubletree (T)\right)  \cdot (e^{-\frac{a}{2}}\alpha(\nu^{\frac{1}{2}}))=u^{n-3}\kappa \alpha Z^u \doubletree c(T).$$

}

{\em Square (6).}The deletions and punctures occur on different strands, as shown in Figure~\ref{fig:456}, hence these operations commute commute. One detail to note is that when a 
tube strand is deleted at a ``tube-and-string'' vertex, all that is left is a string (as in the case of puncturing the tube at a tube-string vertex, see Figure~\ref{fig:punctures}). In summary:
$$ \left(d^3 p^{n-2}u^{n-1} \kappa \alpha Z^u \doubletree (T)\right)  \cdot (e^{-\frac{a}{2}}\alpha(\nu^{\frac{1}{2}}))=p^{n-2}u^{n-3}\kappa \alpha Z^u \doubletree c(T).$$


\parpic[l]{\input figs/ContractCommute7.pstex_t }
{\em Pentagon (7).} The pentagon (7) is shown on the left. This is the most delicate part of the proof. We first show that -- for the specific input of $p^{n-2}u^{n-1}\kappa \alpha Z^u \doubletree(T)$ -- the pentagon (7) commutes up to a single possible multiplicative error on the contracted (u-shaped) strand, and later prove that this error is necessarily 1.


To begin, a better understanding of the arrow diagram  $p^{n-2}u^{n-1}\kappa \alpha Z^u \doubletree(T)$ in the top left corner is necessary. All of the operations performed on $T$, with the exception of $Z^{u}$, are ``easy'' in the sense that we have a complete understanding of their effect. $Z^{u}$ is ``hard'', but we can compute the relevant part of its value using the finite generation of $\sKTG$ (\cite[Proposition 4.13]{Bar-NatanDancso:WKO2}). The computation is shown in Figures \ref{fig:Square7Comm} and \ref{fig:Square7Comm2} and their captions. 

\begin{figure}
\input figs/Square7Comm.pstex_t
\caption{Computing the top left corner of Square 7, Step 1: $\doubletree(T)$  can be expressed as the $\sKTG$ denoted 
$S$ inserted into the $\sKTG$ denoted $A$, followed by unzips, as shown. $Z^{u}$ respects insertions, hence computing $Z^{u}(A)$ determines
the value of $Z^{u}(\doubletree(T))$ outside of $S$.
}
\label{fig:Square7Comm}
\end{figure}

\begin{figure}
\input figs/Square7Comm2.pstex_t
\caption{Computing the top left corner of Square 7, Step 2: computing $Z^{u}(A)$. The $\sKTG$ $A$ can be obtained by inserting the {\em buckle} $\sKTG$ twice
into a simpler $\sKTG$, and unzipping, as shown on the left. The value of the buckle was computed in Figure \ref{fig:BuckleBraid}. Using this value---denoted $\beta^{u}$---and the algorithm in \cite[Section 5.2]{Bar-NatanDancso:WKO2}, one computes $Z^{u}(A)$. The result is denoted $D$ and
shown on the right.}
\label{fig:Square7Comm2}
\end{figure}


In summary,  $Z^u (\doubletree(T))$ is given by inserting $Z^u(A)$ into the chord diagram $D$ of Figure \ref{fig:Square7Comm2}. Now we need to analyze what happens when one applies $\alpha$, the cap attachment, unzips and punctures to this value: this is an exercise similar to what has been done for Lemma~\ref{lem:xiofstrand}. The result is shown in Figure~\ref{fig:Square7Big}, and explained below.
First note that the $n$ value in $D$ of Figure~\ref{fig:Square7Comm2} cancels after punctures by the tail-invariance property (Figure~\ref{fig:Invariance}), as in the last paragraph of the proof of Lemma~\ref{lem:Compatibility}; so does
the bottom $\Phi^{-1}$ in $D$ , by Fact~\eqref{it:Concat} of Lemma~\ref{lem:PhiFacts}. These components are not shown in 
Figure~\ref{fig:Square7Big}.

\begin{figure}
\input figs/Square7Big.pstex_t
\caption{The more detailed picture of the pentagon 7.
}
\label{fig:Square7Big}
\end{figure}


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

Given an arrow `$a$' ending on strand `$e$', unzipping $e$ produces 
a sum of two arrows $a_{1}+a_{2}$: one ending on each daughter strand. Reversing the orientation of the first daughter strand
gives $-a_{1}+a_{2}$. Contracting the two daughter strands to form a U-shape  identifies $a_{1}$
and $a_{2}$, making $a_1-a_2$ vanish.


\begin{figure}
\input figs/KillAnArrow.pstex_t
\caption{Unzip, switch orientation and connect kills arrows on the strand.}
\label{fig:KillAnArrow}
\end{figure}

This is exactly what happens to the arrow diagram $S_2\alpha u(b)$ (just under the tangle $T$ in Figure~\ref{fig:Square7Big}) after the edge deletions or contraction, hence this component cancels it in both directions of the pentagon (7). 

The bottom arrow on the left is the Sorting Isomorphism $\varphi$ of Lemma~\ref{lem:CapString}.

On the other hand, working from the top left corner of the pentagon (7) to the right: from Section~\ref{subsec:AETFormula}, using the strand numbering convention of Figure~\ref{fig:NumberingK}, we have 
$$p_1p_3\beta^w = \Phi^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)}) \cdot e^{a_{23}/2} \cdot \Phi(a_{23},a_{43}).$$
This is shown in the enlarged rectangle at the top right corner of Figure~\ref{fig:Square7Big}. The same is true for $pu(\beta^w)$ at the bottom, except with more unzips.

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

\begin{figure}
\input figs/puBuckle.pstex_t
\caption{The cancellation of $\varphi(pu(\beta^w))$.} \label{fig:puBuckle}
\end{figure}


Of the top $\varphi(p(\beta^{w}))$ shown in the enlarged rectangle, the $\Phi^{-1}$ component cancels by the same contraction argument; only the exponential and the $\Phi$ component remains. The arrow in the exponent of $e^{{a/2}}$ switches sign due to the reversed orientation: this is the $e^{{-a/2}}$ component at the bottom of the pentagon in Figure~\ref{fig:Square7Big}. After contraction, the $\Phi$ component gives rise to a ``local'' arrow diagram on a single strand
shown in Figure~\ref{fig:mu} and denoted $\mu$. 

\begin{figure}
\input{figs/Mu.pstex_t}
\caption{The $\Phi$ component of $\varphi(p(\beta^{w}))$ after contraction.}
\label{fig:mu}
\end{figure}

In summary, we see that the pentagon (7) commutes if and only if $\mu=\alpha(\nu)$, and otherwise commutes up to a localised error on the contracted strand, of value $\alpha(\nu)^{-1}\mu$.

\parpic[l]{\input figs/ContractCommute8.pstex_t }
{\em Square (8).} Finally, for square (8) we need to check that $C^{-1}S(C^{-1})=\alpha(\nu^{-1/2})$. Note that the orientation switch negates odd wheels and preserves even wheels, therefore in $C^{-1}S(C^{-1})$ the odd part of $C^{-1}$ cancels, and $C^{-1}S(C^{-1})=C_0^{-2}=\alpha(\nu^{-1/2})$ by Corollary~\ref{cor:CapValue}. This verifies Square (8).


\vspace{14 mm}
 
We have therefore shown that $Z^{w}$ commutes with contraction up to an error $\alpha(\nu)^{-1}\mu$ on the contracted strand.
It remains to show that this error is $1$. This follows from the facts that $Z^{w}(\uparrow)=1$, and that $Z^{w}$ commutes with 
disjoint unions: 
$$1=c(Z^{w}( \uparrow\downarrow))= \alpha(\nu)^{-1}\mu Z^{w}(c(\uparrow\downarrow))=\alpha(\nu)^{-1}\mu\cdot Z^w(\curvearrowright)=\alpha(\nu)^{-1}\mu.$$
This completes the proof. \qed
\end{proof}

\medskip
Note that as a side result we have proven the following curious fact about associators:
\begin{proposition}
For any $\Phi$ horizontal chord associator, $\mu$ defined from $\Phi$ as in Figure~\ref{fig:mu}, and $\nu$ the Kontsevich integral of the unknot, $\mu=\alpha(\nu)$ in $\calA^{sw}$. \qed 

\end{proposition}

 The value $V=Z^w_{\xi}(\pvertex)$ differs, on first glance, from the value $V_\beta$ constructed in Section~\ref{subsec:Part1}. In the next lemma we show that in fact $V=V_\beta$: this serves both as a reality check, and as a technical tool for showing -- in Theorem~\ref{thm:ZwExpansion} -- that $Z^w_\xi$ satisfies the Cap equation. 

\begin{lemma}\label{lem:TwoConstructions}
The two vertex values from the buckle and the double tree constructions coincide: $V_\beta=V$.
\end{lemma}

\begin{proof}
Notice that $\doubletree(\FlippedYGraph)$ -- as in Figure~\ref{fig:DTandBuckle} -- can be obtained from simpler $\sKTG$s by inserting the buckle $B^u$ into $\doubletree(\curvearrowright)$  followed by an unzip. We computed $\xi(\curvearrowright)$ in Lemma~\ref{lem:xiofstrand}. Since $Z^u$ is compatible with insertions and unzips, $Z^u(\FlippedYGraph)$ can be computed by inserting the buckle value $\beta^u$ (computed in Section~\ref{subsec:AETFormula}) into $Z^u(\doubletree(\FlippedYGraph))$. 

\begin{figure}
\resizebox{\linewidth}{!}{\input figs/DTandBuckle.pstex_t}
\caption{
  On the left we show how to obtain $\doubletree(\FlippedYGraph)$  via
  an unzip from $B^u$ insterted into $\doubletree(\curvearrowright)$.
  From this we compute $\xi(\FlippedYGraph)$, and finally $V$ on the right.
} \label{fig:DTandBuckle}
\end{figure}

To compute $\xi(\FlippedYGraph)$ we then apply $\alpha$ and the cap, unzip and puncture operations. 
Because $\beta^w=\alpha(\beta^u)$ is local (is confined to the skeleton of the inserted $B^u$ graph) and is in the image of $\alpha$, all other arrow endongs commute with it by the head- and tail-invariance properties of $\calA^{sw}$ (Figure~\ref{fig:Invariance}). Therefore, all of the cancellations in the computation of $\xi(\curvearrowright)$ still occur, and $\xi(\FlippedYGraph)$ is as shown in Figure~\ref{fig:DTandBuckle}. Writing this in $\calA^{sw}(\uparrow_2)$ we obtain $\varphi(p_1p_3(\beta^w)) u(\alpha(\nu^{1/2}))$.

To obtain $V=Z^w_\xi(\pvertex)$, one multiplies $\xi(\FlippedYGraph)$ at each end by $C^{-1}$ or $S(C^{-1})$ depending on orientation, as shown in Figure~\ref{fig:DTandBuckle} on the right. Thus, $$V=C^{-1}_1C^{-1}_2\varphi(p_1p_3\beta^w)u(\alpha\nu^{1/2})u(S(C^{-1}).$$

On the other hand, from Section~\ref{subsec:AETFormula}, we have:  
$$V^\beta=C_1^{-1}C_2^{-1}\varphi(p_1p_3\beta^w)u(C).$$ 

Thus, we need to show that:
$$C^{-1}_1C^{-1}_2\varphi(p_1p_3\beta^w)u(\alpha\nu^{1/2})u(S(C^{-1})=C_1^{-1}C_2^{-1}\varphi(p_1p_3\beta^w)u(C),$$
in $\calA^{sw}(\uparrow_2)$.
Multiplying with $\left(\varphi(p_1p_3\beta^w))\right)^{-1}C_1C_2$ on the left (bottom) and by $u(S(C))$ on the right (top), this simplifies:
$$u(\alpha\nu^{1/2})=u(C)u(S(C)).$$
Since unzips commute with orientation switches, it is sufficient to prove that $$CS(C)=\alpha(\nu^{1/2}).$$ Recall that in $CS(C)$ all odd wheels cancel, hence $CS(C)=(C_0)^2$, where $C_0$ denotes the even part of $C$. Indeed, by Corollary~\ref{cor:CapValue}, $C_0^2=\alpha(\nu^{1/2})$. 
\qed\end{proof}

\begin{corollary}\label{cor:SameResult} The ``buckle'' amd ``double tree'' constructions lead to the same result, that is,
$Z^w_\xi=Z^w_\beta.$
\end{corollary}

\begin{proof}
Since any homomorphic expansion $Z^w$ of $\wTFe$ is uniquely determined by $Z^w(\pvertex)$, this is immediate from Lemma~\ref{lem:TwoConstructions}. \qed
\end{proof}


\medskip

The next lemma implies that $V$ satisfies the Unitarity \eqref{eq:U} equation:

\begin{lemma}\label{lem:Unzip}
The map $Z^w_\xi$ commutes with strand unzips in $\uTT$. 
\end{lemma}

\begin{proof}
We first note that in $\uTT$ unzip is only defined for internal edges, that is, edges which end in a vertex\footnote{In fact, there are further restrictions, eg the two vertices must be of opposite signs, but this is not important for the proof.} at both ends.  By construction, $Z^w_\xi$ is a composition of several maps. We show that edge unzips commute with every one of these, hence with $Z^w_\xi$.

The $\doubletree$ map involves only the tangle ends, which are unchanged by unzipping an internal strand, hence these operations commute.
The homomorphic expansion $Z^u$ commutes with edge unzips, as shown in \cite[Section 4.6]{Bar-NatanDancso:WKO2}.
The map $\alpha$ commutes with edge unzips by definition.
Cap attachments, cap unzips and the isomorphism $\varphi$ commute with internal unzips for the same reason as $\doubletree$, because they are performed at the tangle ends while unzip is internal.
This completes the proof.\qed
\end{proof}

Finally, the following Theorem completes the proof of part Part (3) of the Main Theorem~\ref{thm:main}:

\begin{theorem}\label{thm:ZwExpansion}
The map $Z^w_\xi: \wTFe \to \calA^w$ is the unique homomorphic expansion of $\wTFe$, which is compatible with $Z^u$ in the sense of the commutative diagram~\eqref{eq:MainDiag}.
\end{theorem}

\begin{proof}
By Proposition~\ref{lem:Compatibility}, $Z^w_\xi$ is compatible with $Z^u$. 
By Part (1) (Section~\ref{subsec:Part1}) and Corollary~\ref{cor:SameResult}, $Z^w_\xi=Z^w_\beta$ and uniquely determined by $Z^u$.

To show that $Z^w_\xi$ is a homomorphic expansion, by Fact~\ref{fact:EquationsForZ} and Theorem~\ref{thm:ExtendRestrict}, one only needs to verify that it satisfies the \eqref{eq:R4}, Unitarity \eqref{eq:U} and Cap \eqref{eq:C} equations of Fact~\ref{fact:EquationsForZ}. Of these, $R4$ follows from the fact that $Z^w_\xi$ is a planar algebra map, Theorem~\ref{thm:PAMap}. The Unitarity equation \eqref{eq:U} is equivalent to the statement that $Z^w_\xi$ commutes with strand unzips \cite[Section 4.3]{Bar-NatanDancso:WKO2}, and hence it is satisfied by Lemma~\ref{lem:Unzip}.

This leaves the Cap Equation, \qed

\end{proof}
 
%\section{Twist equation and symmetric solutions of the KV equations}



%-------------------------------------------------------------------------------------------------------



\appendix \section{The Alekseev--Enriquez--Torossian formula}\label{app:AET}


 
This appendix is mainly for readers familiar with the Alekseev--Enriquez--Torossian formula for Kashiwara--Vergne solutions in terms of Drinfel'd associators \cite{AlekseevEnriquezTorossian:ExplicitSolutions}.


For a quick re-cap of \cite{AlekseevEnriquezTorossian:ExplicitSolutions} notions, let $\lie_{2}$ denote the degree completed free Lie algebra on two generators $x$ and $y$. Let $\tder_{2}$ denote {\em tangential derivations} of this Lie algebra, that is, derivations $d$ with the property that $d(x)=[x,a_1]$ and $d(y)=[y,a_2]$, where $a_1, a_2 \in \lie_{2}$. Let $\TAut_{2}:= \exp(\tder_{2})$ denote the group of
{\em tangential automorphisms} of $\lie_{2}$. 

There is a linear ``interpretation map'' (not a map of Lie algebras) $\theta: \lie_{2}^{2} \to \tder_{2}$, sending a pair $(a_{1},a_{2})$ to the derivation $d$ given by $d(x)=[x,a_1], d(y)=[y,a_2]$. The kernel of this map consists only of pairs of the form $(\alpha x, \beta y)$ for $\alpha, \beta$ constants. A one-sided inverse $\eta: \tder_{2}\to \lie_{2}^{2}$ which sends a tangential derivation to a pair whose first component has no linear $x$ term and second component has no $y$ term. We denote the exponential of $\theta$ by $\Theta:  \calU(\lie_2^2)_{exp} \to \TAut_{2}$. For an element $(e^{\lambda_1}, e^{\lambda_2}) \in \calU(\lie_2^2)_{exp}$,
we have $G=\Theta((e^{\lambda_1},e^{\lambda_2}))\in \TAut_2$ given by $G(x)=e^{-\lambda_1} x e^{\lambda_1} $, and 
$G(y)=e^{-\lambda_2} y e^{\lambda_2} $. Just as $\theta$ is not a Lie algebra map, $\Theta$ is not a group homomorphism: composition in $\TAut_2$
is not given by piecewise multiplication of the conjugators. However, $\theta$ and $\Theta$ present a convenient way to denote tangential derivations and tangential automorphisms as pairs in $\lie_2^2$ and $(\exp(\lie_2))^2$, respectively.


Given a Drinfel'd associator $\Phi$, \cite[Theorem 4]{AlekseevEnriquezTorossian:ExplicitSolutions} gives an explicit formula for a Kashiwara-Vergne solution $F$ as follows\footnote{There are some notational differences between
\cite{AlekseevTorossian:KashiwaraVergne} and \cite{AlekseevEnriquezTorossian:ExplicitSolutions}.
 There are sign differences between the formula (\ref{eq:AETEx}) and \cite{AlekseevEnriquezTorossian:ExplicitSolutions} due to notational misalignment, for example our $\Phi$ is \cite{AlekseevEnriquezTorossian:ExplicitSolutions}'s $\Phi^{-1}$. Our notation is consistent with the other papers in
this series and the formulas are computationally verified in \cite{Bar-Natan:WKO4}.}:
\begin{equation}\label{eq:AETEx}
F=(\Phi^{-1}(x,-x-y),e^{-(x+y)/2}\Phi(-x-y, y)e^{y/2}).
\end{equation}
In other words, $$F(x)=(\Phi^{-1}(x,-x-y))^{-1}x\Phi^{-1}(x,-x-y),$$ and similarly $$F(y)=(e^{-(x+y)/2}\Phi(-x-y,y)e^{y/2})^{-1}ye^{-(x+y)/2}\Phi(-x-y,y)e^{y/2}.$$
The point of this Appendix is to show that this formula agrees with our Part (2) of the Main Theorem~\ref{thm:main}, via the correspondences between Drinfel'd associators and expansions $Z^u$ \cite[Proposition 3.4]{Bar-Natan:GT1} and between KV-solutions and expansions $Z^w$ \cite[Theorem 4.9]{Bar-NatanDancso:WKO2}.

We first relate $\calA^{sw}(\uparrow_2)$ (where $V$ lives) to $\TAut_2$ (where $F$ lives). A Lie word in $x$ and $y$ can be represented by a binary tree with cyclically oriented vertices, and edges oriented towards a single ``head'': the leaves (tails) are labeled by the letters $x$ and $y$ and the vertices represent brackets. For details see  \cite[Theorem 3.16]{Bar-NatanDancso:WKO2} and the discussion following it. There is a {\em tree attaching} map $l: \tder_{2} \to \calP^{{sw}}(\uparrow_{2})$, where $\calP^{sw}$ denotes the primitive elements of $\calA^{{sw}}$, as follows. Given $d\in \tder_2$, we have $\eta(d)=(a_1,a_2)\in (\lie_2)^2$. Represent the components of $\eta(d)$ by binary trees, and label the heads with $x$ for Lie words coming from $a_1$, and $y$ for $a_2$. Then, attach all $x$-labeled leaves to strand $1$, $y$-labeled leaves to strand $2$, and the head of each tree below all tails. The order of tails is irrelevant by the TC relation. 

Conversely, elements of  $\calP^{{sw}}(\uparrow_{2})$ act as tangential derivations on $\lie_{2}$ \cite[Proposition 3.19]{Bar-NatanDancso:WKO2}. The action is by adding a third strand and embedding $\lie_2 \hookrightarrow \calA(\uparrow_2)$ as $x \mapsto a_{13}$ and $y\mapsto a_{23}$. Then $\calP^{{sw}}(\uparrow_{2})$ embedded in $\calP^{{sw}}(\uparrow_{3})$ on the first two strands acts on $\lie_2 $ by the adjoint action (taking commutators).  Wheels act trivially, and thus one obtains a homomorphism $\delta: \calP^{tree}(\uparrow_{2})\to \tder_{2}$, whose kernel consists of short arrows on either strand. The map $\delta$ is a one-sided inverse to $l$, namely, $\delta \circ l=\operatorname{Id}_{\tder_{2}}$.  


Extending $\delta$ to exponentials 
gives a group homomorphism $\Delta: \calA^w(\uparrow_2)_{exp} \to \TAut_2$, where $ \calA^w(\uparrow_2)_{exp}$ denotes the group-like part of $\calA^{{sw}}$.
For $D\in \calA^w(\uparrow_2)_{exp}$, the map $\Delta$ can be described diagrammatically as follows. Add an extra (third) strand, and embed $\lie_2 \hookrightarrow \calA(\uparrow_2)$ as before. Then $D$, embedded in $\calA(\uparrow_3)$ on the first two strands, acts on $\lie_2$ by conjugation; this defines $\Delta(D)$. 

We construct a map $L$ which completes a commutative triangle as in  Figure~\ref{fig:ATInterpretation}. At the level of primitives, the map $l \circ \theta $ has the property that $\delta \circ (l \circ \theta) = \theta$.
Extend this to the (completed) enveloping algebra $\hat{\calU}(\lie_{2}^{2})$ as follows. An element of $\hat{\calU}(\lie_{2}^{2})$ is an (infinite) linear combination of products of Lie words. As with $l$, represent each Lie word as a labeled tree, but then attach the products of  
these labeled trees to the two strands by attaching {\em all heads} below {\em all tails}. The order of tails doesn't matter, the order of heads is in the order in which the words are multiplied. Call this map $L$, and note that $L$ is {\em not} an algebra homomorphism: it does not respect multiplication in $\hat{\calU}(\lie_{2}^{2})$. However, the restriction of $L$ to the group-like part $\hat{ \calU}(\lie_2^2)_{exp}$, also denoted $L$ fits into a commutative triangle $\Theta=\Delta \circ L$. Note that $L \neq e^l$.


\begin{figure}
$ \xymatrix{
\hat{ \calU}(\lie_2^2)  \ar[dd]^{L} 
 &\hat{ \calU}(\lie_2^2)_{exp}  \ar@{_{(}->}[l]  \ar[dd]^{L} \ar[dr]^{\Theta}
 \\ & & \TAut_2  \\
 \calA^{sw}(\uparrow_2) 
 & \calA^{sw}(\uparrow_2)_{exp}  \ar@{_{(}->}[l] \ar[ur]^{\Delta}
 }$
 \caption{The connection between $\calA^{{sw}}(\uparrow_{2})$ and $\TAut_{2}$.}\label{fig:ATInterpretation}
\end{figure}


Let $Z^u$ be an expansion of $\sKTG$ given by the Drinfel'd associator $\Phi$. Let $V$ be the $Z^{w}$-value of the vertex for the unique homomorphic expansion $Z^{w}$ compatible with $Z^u$. Then $F=\Delta(V)$ 
is a solution to the Kashiwara-Vergne problem  \cite{Bar-NatanDancso:WKO2}.
Our goal is to relate the $\Theta$-image of the \cite{AlekseevEnriquezTorossian:ExplicitSolutions} formula $\Theta(\Phi^{-1}(x,-x-y),e^{-(x+y)/2}\Phi(-x-y, y)e^{y/2})$ to $\Delta(V)$. Note that wheels are in the kernel of $\Delta$ and the \cite{AlekseevEnriquezTorossian:ExplicitSolutions} formula only concerns the {\em tree} component $V^{tree}$.

 
The accomplish this, we need to compute  $V^{tree}=\varphi(\Phi^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)}) \cdot e^{a_{23}/2} \cdot \Phi(a_{23},a_{43}))$ more explicitly, as shown in Figure~\ref{fig:ValueV}, and explained in the caption.  
The first strand of $\calA^{sw}(\uparrow_2)$ joins strands 1 and 2 in a vertex, and the second strand of $\calA^w(\uparrow_2)$
joins strands 3 and 4. Strands 1 and 3 are punctured and strands 2 and 4 are capped. Let us call the two strands of $\calA^w(\uparrow_2)$ strand $I$ and strand $II$ to avoid confusion.
Recall from the construction of $\varphi$ that one first slides arrow tails from the capped strands ``up'' through the vertices, then slides all the heads up from the punctured 
strands 1 and 3. Thus one obtains an element of $\calA^w(\uparrow_2)$ in which all arrow heads are below all tails on both strands. \begin{figure}
 \input figs/ValueV.pstex_t
 \caption{To compute $\varphi(\Phi^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)}) \cdot e^{a_{23}/2} \cdot \Phi(a_{23},a_{43}))$ we switch to a {\em placement notation} in which we mark on each skeleton strand the elements that have arrows ending on it. For this purpose we denote $\Phi^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)})=:\psi$ and $\Phi(a_{23},a_{43})=:\chi$. 
}\label{fig:ValueV}
\end{figure}




Now we are ready to compute how $\pi(V)\in \calA^w(\uparrow_2)$ acts on the generator $x$ of $\lie_2$ and match this to the formula \ref{eq:AETEx}. 
Recall the value of $\pi(V)$ shown in Figure~\ref{fig:ValueV}. The generator $x$ is represented by an arrow from the first strand to the added third
strand, and the action is by conjugation, as shown in Figure~\ref{fig:ActOnx}. To compute this, one commutes the tail of $x$
to the top of the strand across $\pi(V)$ using $\aSTU$ relations, thereby $\pi(V)$ and $\pi(V)^{-1}$ cancel, and the result 
of the action remains. Observe that due to the $TC$ relation, only arrows with heads on strand I act nontrivially on $x$, in other words only $\psi_{1}$ matters, which came from $\Phi^{-1}(a_{2(13), -a_{2(13)}-a_{4(13)}})$. The arrows $a_{23}$ and $a_{43}$ act trivially on $x$, so, more simply stated, the action on $x$ is by 
$\varphi(\Phi^{-1}(a_{21},-a_{21}-a_{41}))$. Note that $L(\Phi^{-1}(x,-x-y),0)=\varphi(\Phi^{-1}(a_{21},-a_{21}-a_{41}))$, so Theorem~\ref{thm:main} agrees with Formula~(\ref{eq:AETEx}) in the first component.


\begin{figure}
 \input figs/ActOnx.pstex_t
 \caption{The action of $\pi(V)$ on the generator $x$ of $\lie_2$.}\label{fig:ActOnx}
\end{figure}

One can proceed similarly for the second component: the action on $y$ is by $$\varphi(\Phi^{-1}(a_{23}, -a_{23}-a_{43})e^{a_{23}}\Phi(a_{23},a_{43}))=L(0,\Phi^{-1}(x, -x-y)e^{x/2}\Phi(x,y)).$$ While this does not match the second component of Formula~(\ref{eq:AETEx}), it only differs from it by a hexagon relation. Alternatively, note that one can obtain the second component of the Formula~(\ref{eq:AETEx}) ``on the nose'' by starting from an equivalent (isotopic) expression\footnote{We thank Karene Chu for this idea.} of $\beta^{b}$, as shown in Figure~\ref{fig:BuckleBraid2}. This completes the proof. \qed

\begin{figure}
 \input{figs/BuckleBraid2.pstex_t}
 \caption{A different expression of $\beta^b$.}\label{fig:BuckleBraid2}
\end{figure}

\smallskip
A sequel paper \cite{Bar-Natan:WKO4} verifies the results in this appendix by explicit computations in low degrees.


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