\draftcut
\section{Some Computations} \label{sec:Computations}

\subsection{Tangle Invariants} \label{subsec:TangleInvariants}

\subsubsection{The General Framework}
Recall from~\cite{WKO2} that the assignment
$\glosm{Zw}{Z^w}\colon\glosm{overcrossing}{\overcrossing} \mapsto
\exp(\glosm{rightarrowdiagram}{\rightarrowdiagram})
\glosm{virtualcrossing}{\virtualcrossing}$ defined on
$S$-component tangles and taking values in $\calA_{\exp}^w(S)$
(where $\rightarrowdiagram$ denotes an arrow connecting the upper strand
to the lower strand and exponentiation is in a formal sense) defines an
invariant of tangles with values in $\calA_{\exp}^w(S)$. We'd
like to compute $Z^w$ (more precisely, its logarithm), in as much as
possible, using both the $\TW_l(S)$-valued \cite{AT}-presentation $E_l$
or using the $\TW_s(S)$-valued factored presentation $E_f$ (recall
Figure~\ref{fig:diagram}).

We let $\glosm{Rl}{R_l^+}(a,b)$ and $\glosm{Rs}{R_s^+}(a,b)$ denote the value
$\glosm{Rab}{R(a,b)} = Z^w\left(\underset{a\ b}{\overcrossing}\right)$
of the positive crossing in $\TW_l$ and $\TW_s$, respectively, and
similarly, let $R_l^-(a,b)$ and $R_s^-(a,b)$ denote the value
$R^{-1}(a,b)=Z^w\left(\underset{b\
a}{\glosm{undercrossing}{\undercrossing}}\right)$ of the negative crossing
in $\TW_l$ and $\TW_s$, respectively (for both signs we label the upper
strand $a$ and the lower strand $b$). That is,
\[
  Z^w\left(\underset{a\ b}{\overcrossing}\right)
    = R^+_l(a,b)\act E_l = R^+_s(a,b)\act E_s
\qquad\text{and}\qquad
  Z^w\left(\underset{b\ a}{\undercrossing}\right)
    = R^-_l(a,b)\act E_l = R^-_s(a,b)\act E_s.
\]
One may easily verify that $R^\pm_{l,s}(a,b)=(a\to 0,\,b\to\pm
a;\,0)_{l,s}$\footnoteC{In computer talk, this is
\mathinclude{RDefs}
}, and it is a simple exercise to verify that $R$ satisfies the Yang-Baxter
/ Reidemeister 3 relation $R_{l,s}^+(1,2)\ast R^+_{l,s}(1,3)\ast
R^+_{l,s}(2,3) = R^+_{l,s}(2,3)\ast R^+_{l,s}(1,3)\ast
R_{l,s}^+(1,2)$\footnoteC{
Indeed, here's a computer verification in $E_l$, to degree 5:

\shortdialoginclude{R3}
}.

\subsubsection{The Knot $8_{17}$ and the Borromean Tangle} In this short
section we evaluate $Z^w$ on the knot $8_{17}$ and on the Borromean tangle,
both shown in Figure~\ref{fig:817AndBorromean}. An expanded version of this
section appears as \cite[Sections \ref{KBH-subsec:Demo1} and
\ref{KBH-subsec:Demo2}]{KBH}.

\begin{figure}
\[ \input{figs/817.pstex_t}\hspace{0.8in}\input{figs/BorromeanTangle.pstex_t} \]
\caption{The knot $8_{17}$ and the Borromean tangle.} \label{fig:817AndBorromean}
\end{figure}

For the 8-crossing knot $8_{17}$ we need to take 8 copies of $R^\pm_s$ with
strands labelled 1 through 16 as in Figure~\ref{fig:817AndBorromean}, and
then stitch strands 1 to 2, 2 to 3, etc\footnoteC{Here it is, to degree 6:

\dialogincludewithlink{817}
}.
This is done using $dm$ operations, and hence we cannot use the $E_l$
presentation.

Similarly for the 6-crossings Borromean tangle we need 6 copies of
$R^\pm_s$ followed by some stitching\footnotemarkC. A colourful
evaluation of the Borromean tangle appears in \cite[Section
\ref{KBH-subsec:Demo2}]{KBH}.

\footnotetextC{To degree 4, we get

\dialogincludewithlink{Borromean}
}

\draftcut
\subsection{Solutions of the Kashiwara-Vergne Equations}
\label{subsec:SolKV} In \cite[Section \ref{2-subsec:wTFo}]{WKO2} we
found that in order to construct a homomorphic expansion $Z^w$ for the
class $\wTFo$ of orientable w-tangled foams, defined there, we need to
find elements $\glosm{V}{V} =
Z^w(\glosm{PlusVertex}{\raisebox{-1mm}{\input{figs/PlusVertex.pstex_t}}}) \in
\calA^w_{\exp}({x,y})$\footnotemarkC\ and $\glosm{Cap}{\Cap}
= Z^w(\glosm{upcap}{\upcap})
\in \calA^w_{\exp}(\upcap_x)$\footnotemarkT~\footnotemarkC\ that are
required to satisfy the three equations in \eqref{eq:HardR4}
and \eqref{eq:UnzipEquations} below. Recall from \cite[Section
\ref{2-subsec:EqWithAT}]{WKO2} that these equations are equivalent
to equations considered by Alekseev and Torossian in~\cite{AT}
(see \cite[Equation \ref{2-eq:ATKVEqns}]{WKO2} and \cite[Section
5.3]{AT}), and that the latter equations were shown in \cite[Section
5.2]{AT} to be equivalent to the Kashiwara-Vergne equations
of~\cite{KashiwaraVergne:Conjecture}.
%
\footnotetextT{$\Cap$ is called $C$ in~\cite{WKO2} and we trust
that the other minor notational differences with~\cite{WKO2} will
cause no difficulty to the reader. Note that $\calA^w(\upcap_S)$ is
$\calA^w(S)$ with $CP$ relations imposed at the tops of the
strands; compare with Section~\ref{sssec:Family}.}
%
\addtocounter{footnoteC}{-1}
\footnotetextC{For computations, we use the $E_s$ presentation
for $V$. As $V$ is presented in $\TW_s(\{x,y\})$, it is of the form
$V=((x\to\alpha,\,y\to\beta);\,\gamma)_s$, where $\alpha,\beta\in\FL(x,y)$
and $\gamma\in\CW(x,y)$, and where the coefficients of $\alpha$, $\beta$,
and $\gamma$, what we call the $\alpha$s, the $\beta$s, and the $\gamma$s,
will be determined later. The first line below sets $\alpha$, $\beta$,
and $\gamma$ to be series with yet-unknown coefficients, and the second
line sets $V$ to be the appropriate combination of $\alpha$, $\beta$,
and $\gamma$:
\mathinclude{VSetup}
(for a technical reason, in computations we use the symbol ${\mathtt
V}_{\mathtt 0}$ to denote $V$).
}
%
\addtocounter{footnoteC}{1}
\footnotetextC{Similarly, $\Cap$ is presented in $\TW_s(x)$. As it is made
only of wheels, its tree part is $0$, or the Lie series \verb$LS[0]$. The
wheels part of $\Cap$ is a series $\kappa\in\CW(x)$ whose
coefficients are the yet-unknown $\kappa$s:
\shortmathinclude{CapSetup}
}

The purpose of this section is to trace through all that at the level
of actual computations. Let us start by recalling from~\cite{WKO2}
the equations for $V$ and for $\Cap$. The first of those is the $R4$
equation \cite[\eqref{2-eq:HardR4}]{WKO2}, $V^{12}\glosm{R}{R}^{(12)3} =
R^{23}R^{13}V^{12}$, coming from the picture
\[ \input{figs/R4ToEquation.pstex_t}. \]
In the language of this paper, and denoting the three strands $x$, $y$, and
$z$, this equation becomes
\begin{equation}\label{eq:HardR4}
  V\ast (R(x,z)\act d\Delta^x_{xy}) = R(y,z)\ast R(x,z)\ast V\footnotemarkC
\end{equation}

The second and the third, ``unitarity'' and the ``cap equation'',
\cite[\eqref{2-eq:unitarity}]{WKO2} and \cite[\eqref{2-eq:CapEqn}]{WKO2},
are the equations
\addtocounter{footnoteC}{-1}
\begin{equation} \label{eq:UnzipEquations}
  V\ast(V\act\dA) = 1\quad\text{in }\calA^w({x,y})
  \quad\text{and}\quad
  V\ast(\Cap\act d\Delta^x_{xy}) = \Cap(\Cap\act d\sigma^x_y)
  \quad\text{in }
  \calA^w(\raisebox{-1mm}{\input{figs/SmallCap.pstex_t}}_{x,y}),\footnotemarkC
\end{equation}
which come from the two unzip operations,
\[ {
  \def\u{{\glosm{unzip}{u}}}
  \def\yu{{\yellowm{u}}}
  \input{figs/Unzips.pstex_t}.
} \]

\footnotetextC{\label{footC:VCapEqns} The three equations in~\eqref{eq:HardR4}
and~\eqref{eq:UnzipEquations} are coded as follows:
\mathinclude{VCapEqns}
}

Solving Equations~\eqref{eq:HardR4} and~\eqref{eq:UnzipEquations} degree
by degree with the initial condition $\alpha=-y/2+\ldots$ we find that
one possible solution, given in the factored presentation, is
\begin{multline*} \small
  V=E_f\left(
      x\to -\frac{\ob{xy}}{24}
        + \frac{7\ob{x\ob{x\ob{xy}}}}{5760} - \frac{7\ob{x\ob{\ob{xy}y}}}{5760}
        + \frac{\ob{\ob{\ob{xy}y}y}}{1440} +\ldots,
        \right. \\
      y\to \frac{\ob{x}}{2} - \frac{\ob{xy}}{12}
        + \frac{\ob{x\ob{x\ob{xy}}}}{5760} - \frac{\ob{x\ob{\ob{xy}y}}}{720}
        + \frac{\ob{\ob{\ob{xy}y}y}}{720} +\ldots; \\
    \left. \vphantom{\frac{\ob{x\ob{x\ob{xy}}}}{720}}
    - \frac{\wideparen{xy}}{48} + \frac{\wideparen{xxxy}}{2880}
      + \frac{\wideparen{xxyy}}{2880} + \frac{\wideparen{xyxy}}{5760}
      + \frac{\wideparen{xyyy}}{2880} + \ldots
  \right)_{\!\!s},
\end{multline*}
and $\small\Cap = - \wideparen{xx}/96 + \wideparen{xxxx}/11,520
- \wideparen{xxxxxx}/725,760 + \ldots$\footnoteC{We set the initial
condition for $\alpha$ in degree 1, then
declare that $\alpha$, $\beta$, $\gamma$, and $\kappa$ are the series which
solve equations \verb$R4Eqn$, \verb$UnitarityEqn$, and \verb$CapEqn$, and
then print the values of $V$ and $\kappa$ (note
the $\hbar^{-1}$ that comes with \verb$R4Eqn$ --- it indicates a degree
shift --- \verb$R4Eqn$ in degree $k$ only puts conditions on our unknowns
at degree $k-1$):

\dialogincludewithlink{VCapSolution}

The solutions of~\eqref{eq:HardR4} and~\eqref{eq:UnzipEquations} are
not unique, and hence occasionally \verb$SeriesSolve$ encounters a
coefficient whose value is not determined by the equations. When this
happens its default action is to set the missing coefficient to $0$. In
the computation this happened to the coefficient of $\wideparen{x}$
in $\kappa$ and to the coefficient of $\ob{\ob{xy}y}$ in $\alpha$.
}.
Note that according
to~\cite{WKO3}, $\Cap$ is always $\sum a_n\wideparen{x^n}$, where
$\sum a_n\hbar^n = \frac14\log\left( \frac{\hbar/2}{\sinh\hbar/2}
\right)$\footnoteC{Indeed, the series below matches with the computation
of $\kappa$, above.

\shortdialoginclude{Sinh}
}.

We can also write $V$ in the lower-interlaced presentation:
\begin{multline*} \small
  V=E_l\left(
      x\to -\frac{\ob{xy}}{24} + \frac{\ob{x\ob{xy}}}{96}
        + \frac{\ob{x\ob{x\ob{xy}}}}{2880} - \frac{\ob{x\ob{\ob{xy}y}}}{480}
        + \frac{\ob{\ob{\ob{xy}y}y}}{1440} +\ldots,
        \right. \\
      y\to \frac{\ob{x}}{2} - \frac{\ob{xy}}{12} + \frac{\ob{x\ob{xy}}}{96}
        + \frac{\ob{x\ob{x\ob{xy}}}}{960} - \frac{\ob{x\ob{\ob{xy}y}}}{320}
        + \frac{\ob{\ob{\ob{xy}y}y}}{720} +\ldots;
    \\ \left. \vphantom{\frac{\ob{x\ob{x\ob{xy}}}}{720}}
    - \frac{\wideparen{xy}}{48} + \frac{\wideparen{xxxy}}{2880}
      + \frac{\wideparen{xxyy}}{2880} + \frac{\wideparen{xyxy}}{5760}
      + \frac{\wideparen{xyyy}}{2880} + \ldots
  \right)_{\!\!s},\footnotemarkC
\end{multline*}
($\Cap$ is the same in both presentations).

\footnotetextC{We could re-compute $V$ in $E_l$ by making some simple
modifications to the input lines in~\ref{footC:VCapEqns}, but it is easier
to use our tools and convert between the two presentations:

\shortdialoginclude{LambdaV}
}

Recall from \cite[Section~\ref{2-subsec:EqWithAT}]{WKO2} and from
Comment~\ref{com:ElAT} that the tree part of ``our'' $V$, taken in
the lower-interlaced presentation, is $\log F^{21}$, where $\glosm{F}{F}$ is
the solution of ``generalized KV problem'' of \cite[Section~5.3]{AT}
and where the superscript $21$ means ``interchange the role of $x$
and $y$''. Thus using the notation of \cite{AT} a solution to degree 4 of the
generalized KV problem is\footnotemarkC
\[ \small
  \log F \!=\! \left(\!
    \frac{\ob{y}}{2} \!+\! \frac{\ob{xy}}{12} \!+\! \frac{\ob{\ob{xy}y}}{96}
      \!-\! \frac{\ob{x\ob{x\ob{xy}}}}{720}
      \!+\! \frac{\ob{x\ob{\ob{xy}y}}}{320}
      \!-\!\frac{\ob{\ob{\ob{xy}y}y}}{960},\,
    \frac{\ob{xy}}{24} \!+\! \frac{\ob{\ob{xy}y}}{96}
      \!-\! \frac{\ob{x\ob{x\ob{xy}}}}{1440}
      \!+\! \frac{\ob{x\ob{\ob{xy}y}}}{480}
      \!-\! \frac{\ob{\ob{\ob{xy}y}y}}{2880}
  \right).
\]

\footnotetextC{The more authoritative version, of course, is the one printed
directly by the computer:

\shortdialoginclude{logF}
}

Next, we'd like to compute a solution of the original Kashiwara-Vergne
equations of~\cite{KashiwaraVergne:Conjecture}. These are the two equations
below, written for unknowns $\glosm{fg}{f,g}\in\FL(x,y)$:
\begin{equation} \label{eq:KV1}
  x + y - \log e^ye^x = (1-e^{-\ad x})f + (e^{\ad y}-1)g,
\end{equation}
\begin{equation} \label{eq:KV2}
  \atdiv_xf + \atdiv_y g
  = \frac12\tr_u\left(\left(
    \frac{\ad x}{e^{\ad x}-1} + \frac{\ad x}{e^{\ad x}-1}
    - \frac{\ad \BCH(x,y)}{e^{\ad \BCH(x,y)}-1}
  \right)(u)\right).
\end{equation}

By tracing the definitions of the comparison map $\kappa$ which appears in
\cite[Theorem~5.8]{AT}, we find that a solution $(f,g)$ of the
Kashiwara-Vergne equations can be computed from $\log F$ via the formula
\[ (f,g) = \frac{e^{\ad(\log F)}-1}{\ad(\log F)}(\calE(\log F)), \]
where $\glosm{calE}{\calE}$ denotes the Euler operator, which multiplies every
homogeneous element by its degree. To degree 4, we find\footnotemarkC\ that
\[ \small
  (f,g) \!=\! \left(\!
    \frac{\ob{y}}{2} \!+\! \frac{\ob{xy}}{6} \!+\! \frac{\ob{\ob{xy}y}}{24}
      \!-\! \frac{\ob{x\ob{x\ob{xy}}}}{180}
      \!+\! \frac{\ob{x\ob{\ob{xy}y}}}{80}
      \!+\!\frac{\ob{\ob{\ob{xy}y}y}}{360},\,
    \frac{\ob{xy}}{12} \!+\! \frac{\ob{\ob{xy}y}}{24}
      \!-\! \frac{\ob{x\ob{x\ob{xy}}}}{360}
      \!+\! \frac{\ob{x\ob{\ob{xy}y}}}{120}
      \!+\! \frac{\ob{\ob{\ob{xy}y}y}}{180}
  \right).
\]

\footnotetextC{With higher authority:

\shortdialoginclude{atkv}

We can then verify that $(f,g)$ indeed satisfy Equations~\eqref{eq:KV1}
and~\eqref{eq:KV2}, at least to degree 9:

\dialoginclude{KVTest}

Of course, we could have simply solved Equations~\eqref{eq:KV1}
and~\eqref{eq:KV2} directly:

\dialogincludewithlink{KVDirect}

(To the degree shown, the results are the same. But starting at degree 8
they diverge as the solutions are non-unique.)
}

\draftcut
\subsection{The involution $\tau$ and the Twist Equation} \label{subsec:tau}
Alekseev and Torossian~\cite[Section 8.2]{AT} construct an
involution $\glosm{tau}{\tau}$ on the set $\SolKV$ of solutions of the
Kashiwara-Vergne equations. Phrased using the language
of~\cite{WKO2}, Alekseev and Torossian define a map
$\tau\colon\calA^w(\uparrow_2)\to\calA^w(\uparrow_2)$ by
$\tau(V)\coloneqq R(1,2)V^{21}\Theta^{-1/2}$,
where $\glosm{Theta}{\Theta}^s=e^{st}$ and
$t=\rightarrowdiagram+\leftarrowdiagram\in\calA^w(\uparrow_2)$. They
then prove that $\tau$ restricts to an involution of the set of solutions
Equations~\eqref{eq:HardR4} and~\eqref{eq:UnzipEquations}. It is not known
if $\tau$ is different from the identity; in other words, it is not known
if every $V$ satisfying~\eqref{eq:HardR4} and~\eqref{eq:UnzipEquations}
also satisfies the ``Twist Equation''
\begin{equation} \label{eq:Twist} V=\tau(V). \end{equation}

\Topology In topology, the Twist Equation is essential for the
compatibility between $\glosm{Zu}{Z^u}$ and $Z^w$;
see~\cite[Section~4.7]{WKO2}. So it is not known if ``every $Z^w$ is
compatible with some $Z^u$''.

Below the dark line we verify that to degree 6, ``our'' $V$ satisfies
the Twist Equation~\eqref{eq:Twist}\footnoteC{We define $\Theta${\tt
l[x,y,s]} to be $e^{st}$ in the $E_l$ presentation in a straightforward
manner, then convert it to the $E_s$ presentation, and then print its
value in both the $E_l$ and $E_s$ presentations:

\dialoginclude{Theta}

This done, the computation of $\tau(V_0)$ and the verification that it
is equal to $V_0$ to degree 6 s routine:

\shortdialoginclude{Vtau}
}.

Following that, we reproduce the results of Albert, Harinck, and
Torossian~\cite{AlbertHarinckTorossian:KV78}, who studied the
linearizations
\begin{equation} \label{eq:LinearizedKV}
  [x,A]+[y,B] = 0
  \qquad\text{and}\qquad
  \atdiv_xA+\atdiv_yB = 0
  \qquad\text{with }A,B\in\FL(x,y)
\end{equation}
of Equations~\eqref{eq:KV1} and~\eqref{eq:KV2} (which are
equivalent to~\eqref{eq:HardR4} and~\eqref{eq:UnzipEquations}), and the
linearization of Equation~\eqref{eq:Twist},
\begin{equation} \label{eq:LinearizedTwist}
  A(x,y) = B(y,x).
\end{equation}

We find\footnotemarkC\ that up to degree 16, the dimensions
of the spaces of solutions of~\eqref{eq:LinearizedKV} and of
\eqref{eq:LinearizedKV}$\wedge$\eqref{eq:LinearizedTwist} are the same
and are given by the following table:

\footnotetextC{We solve for series $A$ and $B$
satisfying~\eqref{eq:LinearizedKV}. These equations are linear, so the
printed solution is $0$. Yet we store messages produced by {\tt
LinearSolve} in a stream called {\tt msgs}. As {\tt LinearSolve}
progresses, it outputs messages detailing which coefficients were set in an
arbitrary manner in each degree, and the dimension of the space of
solutions in each degree can be read from that information:

\dialoginclude{Linearized}

Next, we read the stream {\tt msgs}, just to explore its format:

\shortdialoginclude{msgs}

Next we compute $A$ to degree 12, and read only the dimensions information
contained in {\tt msgs}:

\shortdialogincludewithlink{dims}

Finally we do the same, but now adding Equation~\eqref{eq:LinearizedTwist}:

\dialogincludewithlink{dims1}
}

\begin{equation} \label{tab:dims}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\deg A,B$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\
\hline
dimension  & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &  1 &  1 &  2 &  2 &  3 &  3 &  5 \\
\hline
\end{tabular}
\end{equation}

Assuming that every solution of the KV equations to degree $k$
can be extended to a solution at all degrees (and similarly for
KV$\wedge$Twist)\footnoteT{I am not aware that this was ever proven for
KV (and/or KV$\wedge$Twist), yet a similar result holds for Drinfel'd
associators; see~\cite{Drinfeld:QuasiHopf, Drinfeld:GalQQ, Bar-Natan:NAT,
Bar-Natan:Associators}.}, the above table shows the number of degrees of
freedom for the solutions of KV (and/or KV$\wedge$Twist), in each degree.

\draftcut
\subsection{Drinfel'd Associators} \label{subsec:Associators}
It pains me to say so little
about Drinfel'd associators, but this is a computational paper and
everything we need about associators was already said elsewhere; e.g.,
in Drinfel'd's original papers \cite{Drinfeld:QuasiHopf, Drinfeld:GalQQ},
in my \cite{Bar-Natan:NAT, Bar-Natan:Associators}, and in earlier papers in
this series \cite{WKO2,WKO3}. Hence here I will only recall the few things
that are necessary in order to understand the computations below.

Recall that the Drinfel'd-Kohno algebra $\glosm{fraktn}{\frakt_n}$
is the completed graded Lie algebra with degree $1$ generators
$\{\glosm{tij}{t_{ij}} = t_{ji}\colon 1\leq i\neq j\leq n\}$ and relations
$[t_{ij},t_{kl}]=0$ when $i,j,k,l$ are distinct (``locality relations'')
and $[t_{ij}+t_{ik},t_{jk}]=0$ when $i,j,k$ are distinct (``4T
relations'')\footnoteC{We verify these relations, using obvious notation:

\shortdialoginclude{4T}}. For any fixed $2\leq k\leq n$ the $k-1$
elements $\{t_{ik}\colon 1\leq i<k\}$ form a free subalgebra $\FL_{k-1}$
of $\frakt_n$, and $\frakt_n$ is an iterated semi-direct product of
these subalgebras:
\begin{equation} \label{eq:DKSD}
  \frakt_n \cong
  ((\ldots(\FL_1\ltimes\FL_2)\ltimes\ldots)\ltimes\FL_{n-2})\ltimes\FL_{n-1}.
\end{equation}
Hence as a vector space, $\frakt_n$ has a basis with elements ordered
pairs $(k,w)$, where $2\leq k\leq n$ and $w$ is a Lyndon word in the
letters $\{1,\ldots,k-1\}$ (which really stand for
$\{t_{1k},\ldots,t_{k-1,k}\}$)\footnoteC{Hence for example,
$[t_{13},t_{12}]=-[t_{13},t_{23}]$ (the bracket of a generator of $\FL_3$
with the generator of $\FL_2$ is an element of $\FL_3$). In computer speak,
this is

\shortdialoginclude{DKExample}

Note that the head \verb$DK$ represents ``a basis element in a
Drinfel'd-Kohno algebra'', and that the Lyndon word $12$ becomes
$[t_{13},t_{23}]$ when interpreted in $\FL_3\subset\frakt_3$.

We could make the last output a bit friendlier by turning it into a
``Drinfel'd-Kohno Series'' (\verb$DKS$):

\shortdialoginclude{DKSExample}
}. 

The collection $\{\frakt_n\}$ of all Drinfel'd-Kohno algebras forms an
``operad'' (e.g.~\cite{Fresse:OperadsAndGT}). We only need to mention a
part of that structure here: that for any $n$ and $m$, there are many maps
$\frakt_n\to\frakt_m$. Namely, whenever $\left\{s_i\right\}_{i=1}^n$ is a
collection of disjoint subsets of $\{1,\ldots,m\}$ (some of which may be
empty), we have a morphism of Lie algebras
$\Psi\mapsto\Psi^{s_1,\ldots,s_n}$ mapping $\frakt_n$ to $\frakt_m$,
and defined by its values on the generators of $\frakt_n$ as follows:
\[ \left(t_{ij}\right)^{s_1,\ldots,s_n} \coloneqq
  \sum_{\alpha\in s_i,\,\beta\in s_j} t_{\alpha\beta}.\footnotemarkC
\]

\footnotetextC{As an example we repeat a single evaluation of a map
$\frakt_4\to\frakt_9$ twice. First using a complete and somewhat cumbersome
notation, and then using a shortened notation that works only if all
indices are single-digit:

\shortdialoginclude{sigmaExample}
}

Note also that by regarding elements of $\frakt_n$ as formal
exponentials and using the BCH product each $\frakt_n$ also acquires a
(non-commutative) group structure.\footnoteC{For example, in $\frakt_3$ the
elements $t_{12}$ and $t_{23}$ do not commute, and hence the product
$e^{t_{12}/2}e^{t_{23}/2}$ is messy. Yet by a 4T relation the elements
$t_{12}$ and $(t_{12})^{12,3}=t_{13}+t_{23}$ do commute, and hence the
product $e^{t_{12}/2}\left(e^{t_{12}/2}\right)^{12,3}$ is much simpler:

\shortdialoginclude{BCH4DK}
}
By convention, when we think of $\frakt_n$ as a group, we refer to it as
``$\glosm{exptn}{\exp\frakt_n}$''.

We are finally in position to recall the definition of a
Drinfel'd associator. With $R=e^{t_{12}/2}\in\exp\frakt_2$, a
Drinfel'd associator is an element $\glosm{Phi}{\Phi}\in\exp\frakt_3$ which
satisfies the ``unitarity condition''~\eqref{eq:PhiUnitarity},
the pentagon equation~\eqref{eq:PhiPentagon}, and the hexagon
equations~\eqref{eq:PhiHexagons}:

\begin{align}
  \text{Unitarity}&\colon& \Phi^{321}
    &= \Phi^{-1}, \label{eq:PhiUnitarity} \\
  \pentagon&\colon& \Phi\cdot\Phi^{1,23,4}\cdot\Phi^{2,3,4}
    &= \Phi^{12,3,4}\cdot\Phi^{1,2,34}, \label{eq:PhiPentagon} \\
  \hexagon_\pm&\colon& (R^{\pm 1})^{12,3}
    &= \Phi\cdot(R^{\pm 1})^{2,3}\cdot(\Phi^{-1})^{1,3,2}\cdot
       (R^{\pm 1})^{1,3}\cdot\Phi^{3,1,2}. \label{eq:PhiHexagons}
\end{align}

A surprising result by Furusho~\cite{Furusho:Pentagon} (see
also~\cite{Bar-NatanDancso:Furusho}) states that in the context of
$\exp\frakt_n$ the hexagon equations follow from unitarity and the
pentagon, provided $\Phi$ is initialized to degree $2$ by
$\Phi=\exp\left([t_{13},t_{23}]/24+\text{higher
terms}\right)$.\footnoteC{Here's an associator $\Phi_0$, computed to
degree $6$. The data file~\citeweb{Phi.nb} contains a computation
of an associator to degree 10, higher than was previously
computed~\cite{Bar-Natan:NAT, Brochier:DrinfeldAssociators}.

\dialogincludewithlink{Phi}

To be on the safe side, we verify that $\Phi_0$ satisfies the hexagon
equations to degree $6$:

\dialoginclude{Hexagons}}

\draftcut
%\Needspace{256mm} % ??? \Needspace failure.
\parpic[r]{\input{figs/PhiV.pstex_t}}
\subsection{Associators in $\calA^w$} \label{subsec:wAssociators}
We know from~\cite[Section~1]{AT}
that a certain combination of four copies of $V$ makes a solution
of the pentagon equation, with values in $\tder_3$. In the language
of~\cite{WKO2}, this is the statement that $V$ is the $Z^w$-value
of a vertex, that four vertices can make a tetrahedron, and that the
$Z^w$-value $\glosm{PhiV}{\Phi_V}$ of a tetrahedron is an associator in
$\calA^w$ (see the figure on the right). Specifically,
\[ \Phi_V = (V\act\dA)^{12,3}(V\act\dA)^{1,2}V^{2,3}V^{1,23},\footnotemarkC \]
where we use standard notation: $V^{2,3}$, for example, means ``$V$ with
its $x$ strand renamed $2$ and its $y$ strand renamed $3$'' and $V^{1,23}$
means ``V with its $x$ strand renamed $1$ and its $y$ strand doubled to
become strands $2$ and $3$''. With the language of
Definition~\ref{def:Operations}, this is $V^{2,3}=V\act d\sigma^x_2\act
d\sigma^y_3$ and $V^{1,23} = V\act d\sigma^x_1\act d\Delta^y_{23}$.

\footnotetextC{And here is $\Phi_V$, to degree 4:

\shortdialoginclude{PhiV}}

$\Phi_V$ satisfies the pentagon equation.\footnoteC{Indeed,

\shortdialoginclude{PentPhiV}}
If our $V$ also satisfies the Twist Equation, then $\Phi_V$ also satisfies
the hexagon equations (though we do not test that here). Finally, Alekseev
and Torossian~\cite{AT} prove that if the tree part of $\Phi_V$ is written
as an exponential $\exp(l\phi)$ of an element $\phi$ of $\tder_3$,
then in fact $\phi\in\sder_3$, where as in~\cite{AT},
$\glosm{sder}{\sder}_n$ is the space of ``special derivations in
$\tder_n$'', the derivations which annihilate the sum of all
generators on $\FL_n$\footnotemarkC.

\footnotetextC{We convert $\Phi_V$ to the $E_l$ presentation and take
its first (tree) part and call it $\phi$, and then we verify that
$[x_1,\phi_1] + [x_2,\phi_2] + [x_3,\phi_3] = 0$:

\shortdialoginclude{Phi_is_sder}}

\vskip -2mm
\noindent\parbox[b]{\textwidth-2.0in}{
\Topology The topological meaning of ``$\phi\in\sder_3$'' is that one
may perform a sequence of four $R4$ moves to slide a strand underneath
a tetrahedron, as shown on the right.
}\hfill\input{figs/Phisder.pstex_t}
\vskip 2mm

Recall that there is a map
$\glosm{alpha}{\alpha}\colon\frakt_n\to\calA_{\text{prim}}^w(\uparrow_n)$
(equivalently, $\alpha\colon\calU(\frakt_n)\to\calA^w(\uparrow_n)$),
defined by its values on the generators by sending $t_{ij}$ to a sum of a
single arrow from strand $i$ to strand $j$ plus a single arrow from strand
$j$ to strand $i$: $t_{ij}\mapsto\tensor[_i]{\rightarrowdiagram}{_j}
+ \tensor[_i]{\leftarrowdiagram}{_j}$. Using the map $\alpha$, every
Drinfel'd associator becomes an associator in $\calA^w$.\footnoteC{Indeed,
we define a map {\tt DK2Es} which takes Drinfel'd-Kohno series to elements
of $\calA^w$ given in the $E_s$ presentation by applying the built-in
$\alpha${\tt Map}, adding $0$ wheels, and applying the $E_l$ to $E_s$
conversion $\Gamma$. Applying this map to the Drinfel'd associator
$\Phi_0$ computed before, we get and associators in $\calA^w$:

\shortdialoginclude{DK2Es}

The result matches $\Phi_V$, computed before, to the degree shown. But this
is only because both associators are supported in even degrees, and there's
a unique even associator in $\calA^w$ up to degree $4$. In degree $8$
these two associators diverge.}

\Topology In topology, $\alpha$ is the associated graded of the ``do
nothing'' map $\glosm{uva}{a}$ which maps ordinary knots to virtual knots.
$\horizontalchord\mapsto\rightarrowdiagram + \leftarrowdiagram$ because
$\horizontalchord\sim\doublepoint\sim\overcrossing-\undercrossing \mapsto
(\overcrossing-\virtualcrossing) + (\virtualcrossing-\undercrossing) \sim
\semivirtualover + \semivirtualunder \sim
\rightarrowdiagram + \leftarrowdiagram$. See
\cite[Section~\ref{1-subsubsec:RelWithu}]{WKO1} and
\cite[Section~\ref{2-subsec:sder}]{WKO2}.

\Lie In Lie theory, the existence of $\alpha$ corresponds to the
fact that the invariant metric on $I\frakg=\frakg\ltimes\frakg^\ast$
(represented by an undirected chord) is the sum of the two possible
contractions of a space with its dual in
$(\frakg\ltimes\frakg^\ast)\otimes(\frakg\ltimes\frakg^\ast)$ (the two
arrows).

\AT The~\cite[Proposition~3.11]{AT} version of $\alpha$ is the
map $\frakt_n\to\atsder_n \subset \attder_n$ taking $t_{ij}$ to
$\partial\left(i\to x_j,\, j\to x_i,\, (k\neq i,j)\to 0\right)$.

\draftcut
\subsection{Solving the Kashiwara-Vergne Equations Using a Drinfel'd Associator}
\label{subsec:KVandAssoc} Following~\cite{WKO3} (in a deeper
sense, following~\cite{AlekseevEnriquezTorossian:ExplicitSolutions}),
we know that an element $V$ solving the KV equations \eqref{eq:HardR4}
and \eqref{eq:UnzipEquations} can be computed from a Drinfel'd associator
$\Phi$ by first computing the invariant $\glosm{ZB}{Z_B} = Z^u(B)$ 
of the ``buckle'' $\glosm{B}{B}$, shown below
both as a knotted trivalent graph and as a product of associators,
then puncturing strands 1 and 3 and capping strands 2 and 4 from below,
and then regarding the result in $\calA^w(\uparrow_2)$ by applying an
``Etingof-Kazhdan (EK) isomorphism'':\footnotemarkC
\[
  B = \begin{array}{c}\input{figs/Buckle.pstex_t}\end{array}
  \mapsto Z_B = (\Phi^{-1})^{13,2,4}\Phi^{1,3,2}R^{23}\Phi^{-1}\Phi^{12,3,4}
  \xrightarrow{\text{puncture, cap, EK}} V.
\]

\footnotetextC{We start with a straightforward computation of $Z_B$:

\shortdialoginclude{ZB}

In the $E_s$ presentation, ``puncture'' is $t\eta$. So we puncture strands
1 and 3:

\shortdialoginclude{VfromPhi}

At this point we would normally need to cap and apply EK. But fortunately,
strands 2 and 4 carry no arrow heads (as can be seen in the above output),
so there is no need to cap them and the EK isomorphisms act by doing
nothing. Hence apart from some obvious renaming, the above is already a
solution of the KV equations. It matches with the previously-computed $V$
to degree 4 but diverges from it in degree 8 (not shown here). This is
consistent with the result in~\eqref{tab:dims}, which shows that
non-uniqueness starts only in degree $8$.}

\Needspace{1in}
\parpic[r]{\input{figs/nu.pstex_t}}
Likewise following~\cite{WKO3}, we know that $\Cap=\alpha(\nu^{1/4})$, where
$\glosm{nu}{\nu}$ is the Kontsevich integral of the unknot, or the
inverse of the associator-combination shown on the right and given by
the formula $\alpha(\nu^{-1})=\Phi\act\alpha\act \dS^2\act dm^{32}_2\act
dm^{21}_1$.\footnotemarkC\ (Note that this computation uses the operation
$\dS^a$, which is not easily available in the $E_l$ presentation).

\footnotetextC{Indeed here is $\nu^{-1}$, followed by a verification that
$\nu^{-1}\Cap^4$ is trivial:

\shortdialoginclude{nu}

\shortdialoginclude{nucap4}}

An alternative (yet equivalent) formula for $V$ in terms of $\Phi$
follows~\cite{AlekseevEnriquezTorossian:ExplicitSolutions}
more closely. Indeed according
to~\cite[Theorem~4]{AlekseevEnriquezTorossian:ExplicitSolutions}
and~\cite{WKO3} $V$ generates the tangential automorphism of $\FL(x,y)$
given explicitly by $(x\mapsto F_xxF_x^{-1}, y\mapsto F_yyF_y^{-1})$, where
\begin{equation} \label{eq:AET}
  F=(F_x,F_y)=\left(
    \Phi^{-1}(x,-x-y),\ e^{(x+y)/2}\Phi^{-1}(y,-x-y)e^{-y/2}
  \right)
\end{equation}
(though note that our conventions here agree with the conventions
of~\cite{WKO3} but slightly differ from the conventions
of~\cite{AlekseevEnriquezTorossian:ExplicitSolutions}).

Below the line we verify Equation~\eqref{eq:AET}.\footnoteC{We first have
to rewrite $\Phi$ in terms of $x=t_{12}$ and $y=t_{23}$. To do this we
``3'' term in $\Phi$, the one involving $t_{13}$ and $t_{23}$ in the
factorization~\eqref{eq:DKSD} (it is the only non trivial term), and apply
the appropriate change of variables $t_{13}\to-x-y$, $t_{23}\to y$. It is
smooth sailing afterwards:

\shortdialoginclude{Phi1}

\shortdialoginclude{F}

\shortdialoginclude{FV}}

\draftcut
% The "minipage" here is to prevent a very funny bad interaction of the
% section title with the "computations below" rule.
\noindent\begin{minipage}{\textwidth}
\subsection{A Potential $S_4$ Action on Solutions of KV}
\label{subsec:Trivolution} In~\cite{Bar-NatanDancso:KTG},
Z.~Dancso and I discussed how ``the expansion of a tetrahedron''
can be interpreted as an associator valued in the appropriate
space $\calA^u(\glosm{tetrahedron}{\tetrahedron})\cong\calA^u(\uparrow_3)$ (see
also~\cite{Thurston:Shadow}). The symmetry group of an oriented
tetrahedron is the alternating group $A_4$, and hence $A_4$
acts on the set of all associators in $\calA^u(\uparrow_3)$
(note that while the action of the permutation group $S_3$ on
$\calA^u(\uparrow_3)$ is obvious, its extension to an action of
$S_4$ is non-obvious and is best understood using the isomorphism
$\calA^u(\tetrahedron)\cong\calA^u(\uparrow_3)$). The unitarity
equation~\eqref{eq:PhiUnitarity} means that odd permutations map
associators to objects whose inverses are associators; with some abuse
of language we simply say that ``$S_4$ acts on the set of associators''
(really, it acts on ``associators and inverse-associators''). As there
are bi-directional relations between associators and solutions of the
KV equations, we can expect an action of $S_4$ on the set of solutions
of the KV equations and their inverses.
\end{minipage}

\vskip 1mm
As mathematicians, Z.~Dancso and I only lightly explored this
potential action of $S_4$; we wrote down what we think are the formulas
inherited from the action on associators, but on the formal level, we've
verified almost nothing. Yet computer experiments, described below,
suggest that our formulas are correct and that they have the properties
described below.

\noindent{\bf The first $\bbZ/2$ action} is the involution $\tau$ discussed
in Section~\ref{subsec:tau}. We have nothing further to add.

\noindent{\bf The second $\bbZ/2$ action} is the involution
$\glosm{rho2}{\rho_2}$ of $\calA^w$ which multiplies every degree $d$
element by $(-1)^d$. Solutions $V$ of the KV equations are not invariant
under $\rho_2$. Yet if $V_0$ is the solution computed in this paper
then $V_1\coloneqq R^{-1/2}V_0$ is invariant under $\rho_2$, at least
experimentally. Alternatively, $V_0$ is (experimentally) invariant under
$\rho_2'\coloneqq R\rho_2$.\footnoteC{Indeed,

\dialoginclude{rho2}
}

\noindent{\bf A $\bbZ/3$ action.} For $\xi\in\calA^w({x,y})$
let $\glosm{rho3}{\rho_3}(\xi)\coloneqq \xi\act \dS^y \act d\Delta^y_{yz} \act
dm^{xz}_x \act d\sigma^{xy}_{yx}$, where $d\sigma^{xy}_{yx}$ simply
means ``swap the labels $x$ and $y$''. Then $\rho_3$ is a trivolution
($(\rho_3)^3=1$)\footnoteC{Indeed for a random $\xi_c$,
$\xi_c\act\rho_3\act\rho_3\act\rho_3=\xi_c$:

\dialoginclude{rho3}
}, and a renormalized version of $V_0$, namely $V_2\coloneqq V_0 \ast
\Theta^{-1/4} \ast \exp\left(\frac{\wideparen{x}-\wideparen{y}}{12}\right)
\ast d\Delta^x_{xy}(\Cap^2)$ is, at least experimentally, invariant
under the action of $\rho_3$.\footnoteC{Indeed,

\begin{minipage}{\textwidth}\vskip 1mm\dialoginclude{V2}\end{minipage}}