\draftcut\section{Introduction} Within the previous three
papers in this series~\cite{WKO1}--\cite{WKO3}\footnoteT{Also within
my~\cite{KBH}, and within papers by Alekseev, Enriquez,
and Torossian~\cite{AT,
AlekseevEnriquezTorossian:ExplicitSolutions}, and within
Kashiwara's and Vergne's~\cite{KashiwaraVergne:Conjecture},
and also within many older papers about Drinfel'd associators
(e.g. Drinfel'd's~\cite{Drinfeld:QuasiHopf, Drinfeld:GalQQ} and my
\cite{Bar-Natan:NAT}.} a number of intricate equations written in various
graded spaces related to free Lie algebras and to spaces of cyclic words
were examined in detail, for good reasons that were explained there and
elsewhere. The purpose of this paper is to introduce mathematical tools
(on the upper parts of pages) and computational tools (on the lower
parts of pages, below the bold dividing lines\footnotemarkC) that allow for
the explicit solution of these equations, at least up to a certain degree.
\footnotetextC{
If you are not interested in the actual computations, it is safe to
ignore the parts of pages below the bold dividing lines and restrict to
``strict'' mathematics, which is always above these lines. {\bf Alert.} If
you are interested in the computations, note that the computational
footnotes are sometimes long and crawl across page boundaries. This
footnote is the first example.
The programs described in this paper were written in
Mathematica~\cite{Wolfram:Mathematica} and are available at~\cite{WKO4}.
Before starting with any computations, download the packages
\href{\web/FreeLie.m}{\tt FreeLie.m} and
\href{\web/AwCalculus.m}{\tt AwCalculus.m} and type within Mathematica:
(the interactive Mathematica session demonstrated in this paper is
available as \citeweb{WKO4Session.nb})
\dialoginclude{Initialization} \rule{0in}{6pt}
% For unknown reasons the \rule above prevents a page break here.
The last input (``human'') line above declares that by default we wish the
computer to print series within graded spaces (such as free Lie algebras)
to degree 4. Note that we \lightred{highlight in pink} input lines that
affect later computations.
}
The equations we have in mind arise in other papers and appear throughout
this paper. Yet to help our impatient readers orient themselves,
Figure~\ref{fig:flash} contains a ``flash summary'' of the most important
equations and their topological and algebraic significance.
\begin{figure}
\[
\def\YB{$R^{12}R^{13}R^{23}=R^{23}R^{13}R^{12}$}
\def\RFour{$R^{23}R^{13}V=R^{12,3}$}
\def\RFourAT{{\cite{AT}:
$F(x+y)=\log e^xe^y$
}}
\def\UniCap{$VV^\ast=1;\quad VC^{12}=C^1C^2$}
\def\UniCapAT{{\cite{AT}:
$j(F)\in\im(\tilde{\delta})$
}}
\def\Twist{$\Theta=V^{-1}RV^{21}$}
\def\Pentagon{$\Phi\Phi^{1,23,4}\Phi^{234}=\Phi^{12,3,4}\Phi^{1,2,34}$}
\def\Buckle{$(\Phi^{-1})^{13,2,4}\Phi^{132}R^{23}\Phi^{-1}\Phi^{12,3,4}$}
\input{figs/FlashSummary.pstex_t}
\]
\caption{The most important equations.} \label{fig:flash}
\end{figure}
Why bother? What do limited explicit computations add, given that these
intricate equations are known to be soluble, and given that the conceptual
framework within which these equations make sense is reasonably well
understood~\cite{WKO1}--\cite{WKO3}? My answers are three:
\begin{enumerate}[leftmargin=*,labelindent=0pt]
\item Personally, my belief in what I can't compute decays quite rapidly
as a function of the complexity involved. Even if the overall picture is
clear, the details will surely go wrong, and sooner or later, something
bigger than a detail will go wrong. Even a limited computation may
serve as a wonderful sanity check. In situations such as ours, where
many signs and conventions need to be decided and may well go wrong,
even a low-degree computation increases my personal confidence level by a
great degree. Given computations that work to degree 6 (say), it is hard
to imagine that a detail was missed or that conventions were established
in an inconsistent manner. In fact, if the computer programs are clear
enough and are shown to work, these programs become the authoritative
declarations of the details and conventions.
\item The computational tools introduced here may well be useful in other
contexts where free Lie algebras and/or cyclic words arise.
\item The papers~\cite{WKO1,WKO2} (and likewise~\cite{KBH}) are about
equations, but even more so, about the construction of certain knot and
tangle invariants. With the tools presented here, the invariants of
arbitrary knotted objects of the types studied in~\cite{WKO1,WKO2,KBH}
may be computed.
\end{enumerate}
The equations of~\cite{WKO1}--\cite{WKO3} always involve group-like,
or ``exponential'' elements, and are written in some spaces of ``arrow
diagrams'' that go under the umbrella name $\calA^w$. Hence a crucial
first step is to find convenient presentations for the group-like
elements $\calA^w_{\exp}$ in $\calA^w$-spaces. It turns out that there
are (at least) two such presentations, each with its own advantages and
disadvantages. Hence in Section~\ref{sec:Aw} we recall $\calA^w$ briefly
(\ref{subsec:Aw}), then discuss some free-Lie-algebra preliminaries
(\ref{subsec:FL}), then describe the Alekseev-Torossian-\cite{AT}-inspired
``lower-interlaced'' presentation $E_l$ of $\calA^w_{\exp}$
(\ref{subsec:AT}), then describe the \cite{KBH}-inspired ``factored''
presentation $E_f$ of $\calA^w_{\exp}$ and its stronger precursor
``split'' presentation $E_s$ (\ref{subsec:Ef}), and then describe how
to convert between the two primary presentations (\ref{subsec:Conversion}).
We then present our computations in Section~\ref{sec:Computations}: Some
knot and tangle invariants are computed in
Section~\ref{subsec:TangleInvariants} and solutions of the Kashiwara-Vergne
(KV) equations in Section~\ref{subsec:SolKV}. In Section~\ref{subsec:tau} we
discuss the ``Twist Equation'' and compute dimensions of spaces of
solutions of the linearized KV equations, with and without the Twist
Equation. In Section~\ref{subsec:Associators} we compute a Drinfel'd
associator, in Section~\ref{subsec:wAssociators} we compute associators in
$\calA^w$ starting from a solution of the KV equations, and in
Section~\ref{subsec:KVandAssoc} we show how to compute a solution of KV
from a Drinfel'd associator. The last computational result is in
Section~\ref{subsec:Trivolution}, where we give computational support to
the existence of an action of the symmetric group $S_4$ on the set of
solutions of the Kashiwara-Vergne Equations.
\begin{figure}
\[ \xymatrix@C=0.8in@R=0.8in{
\left\{\TW_l(S)\right\}
\ar@`{p+(16,16),p+(-16,16)}_{\ast,{\gray dm},\ldots}
\ar@/^/[r]^\Gamma
\ar[d]^{E_l}
& \left\{\TW_s(S)\right\}
\ar@`{p+(16,16),p+(-16,16)}_{\ast,\#,dm,\ldots}
\ar@/^/[l]^{\Lambda=\Gamma^{-1}}
\ar@{^{(}->}[r]
\ar[d]^{E_s}
\ar[dl]^(0.4){E_f}
& \left\{\TW_s(H;T)\right\}
\ar@`{p+(16,16),p+(-16,16)}_{\ast,\#,dm,hm,tm,tha,\ldots}
\ar[d]^{E_s}
\\
\left\{\calA^w_{\exp}(S)\right\}
\ar@`{p+(-16,-16),p+(16,-16)}_{\ast,dm,\ldots}
\ar@/^/[r]^{\delta}
& \left\{\calA^w_{\exp}(S;S)\right\}
\ar@`{p+(-16,-16),p+(16,-16)}_{\ast,\#,dm,\ldots}
\ar@/^/[l]^{\delta^{-1}}
\ar@{^{(}->}[r]
& \left\{\calA^w_{\exp}(H;T)\right\}
\ar@`{p+(-16,-16),p+(16,-16)}_{\ast,\#,dm,hm,tm,tha,\ldots}
} \]
\caption{The main spaces and maps appearing in this paper.}
\label{fig:diagram}
\end{figure}
We conclude this introduction with a description of the commutative
diagram in Figure~\ref{fig:diagram} which displays the main
spaces and maps appearing in this paper, as described in detail in
Section~\ref{sec:Aw}. The bottom row of this diagram consists of spaces of
``group-like'' elements inside spaces $\calA^w$ of ``arrow diagrams'';
these are the spaces that have direct knot-theoretic significance. The
top row are spaces of ``trees and wheels'', or more precisely, various
elements of free Lie algebras and various cyclic words. They are the
spaces of ``primitives'' corresponding to the group-like elements at the
bottom, via various ``exponentiation'' maps $E_l$, $E_f$, and $E_s$. In
this paper we study\footnoteC{Or ``implement'', in computer-speak.} the
spaces on the bottom row by means of their presentations by elements in
the top row.
The collection $\left\{\calA^w_{\exp}(S)\right\}$ of spaces
we primarily wish to study (and in which most of the equations of
Figure~\ref{fig:flash} are written) appears on the bottom left. There
are many binary and unary operations acting on the spaces within
$\left\{\calA^w_{\exp}(S)\right\}$ as indicated by the circular
self-arrow appearing there, which is labelled with the most important
of these operations, the binary $\ast$ and the unary $dm$. On the top
left of the diagram are the spaces $\left\{\TW_l(S)\right\}$ of trees
and wheels which represent $\left\{\calA^w_{\exp}(S)\right\}$
via the $E_l$ presentation. The same collection of operations acts here
too, though notice that the operation $dm$ is grayed-out, because we
have no direct implementation for it in $\TW_l$ language.
On the bottom right is a bigger collection of spaces,
$\left\{\calA^w_{\exp}(H;T)\right\}$, which contains as a subset the
collection $\left\{\calA^w_{\exp}(S;S)\right\}$ (bottom middle), which
is isomorphic in a non-trivial manner (via $\delta$ and $\delta^{-1}$) to
$\left\{\calA^w_{\exp}(S)\right\}$. A richer collection of
operations act on $\left\{\calA^w_{\exp}(H;T)\right\}$, and the most
important of those are $\ast$, $\#$, $dm$, $hm$, $tm$, and $tha$.
On the top right is the collection $\left\{\TW_s(H;T)\right\}$ of spaces
of trees and wheels which represent $\left\{\calA^w_{\exp}(H;T)\right\}$
via the $E_s$ presentation. When restricted to $H=T=S$,
this is the collection $\left\{\TW_s(S)\right\}$ representing
$\left\{\calA^w_{\exp}(S;S)\right\}$, and representing our primary
interest $\left\{\calA^w_{\exp}(S)\right\}$ via $E_f$, the
composition of $E_s$ with $\delta^{-1}$.
Note that $\TW_l$ and $\TW_s$ are set-theoretically the same spaces
of trees and wheels. Yet the operations $\ast$, $dm$, etc.\ act on
them in a different manner, and hence they deserve to have different
names\footnoteT{Much as in group theory, a direct product $N\times H$
is set-theoretically the same as a semi-direct product $N\rtimes H$, yet
it is wrong to refer to them by the same name.}. Note also that $\TW_l$
and $\TW_s$ are in fact isomorphic via structure-preserving isomorphisms
(denoted $\Gamma$ and $\Lambda=\Gamma^{-1}$). These isomorphisms are
compositions of the relatively simple-minded $\delta$ and $\delta^{-1}$
with the more complex ``exponentiations'' $E_l$ and $E_s$ and their
inverses. Thus the isomorphisms $\Gamma$ and $\Lambda$ are non-linear
and quite complicated.
\TopLieAT We will occasionally comment on the relationship between
the constructs appearing in this papers and three related topics:
``topology'', or more precisely certain aspects of the theory of
2-knots, ``Lie theory'', or more precisely certain classes of formulas
that make sense in arbitrary finite-dimensional Lie algebras, and
``Alekseev-Torossian'', or more precisely, issues related to the
paper~\cite{AT}. These comments will in general be incomplete and should
be regarded as ``hints for the already initiated'' --- people familiar
with the papers \cite{WKO1, WKO2, WKO3, KBH, AT} will hopefully find
that these comments help to put the current paper in context. These
comments will always be labelled by one (or more) of the three logos at
the head of this paragraph, which correspond, in order, to ``topology'',
``Lie theory'', and ``Alekseev-Torossian''.
\Topology Within the study of simply-knotted (ribbon) 2-knots, or more
precisely w-knotted-objects as they
appear in \cite{WKO1, WKO2, KBH}, the rows of
Figure~\ref{fig:diagram} correspond to the extra row
\[ \xymatrix@C=0.8in{
\left\{\calK^w(S)\right\}
\ar@`{p+(-16,-16),p+(16,-16)}_{\ast,dm,\ldots}
\ar@/^/[r]^{\delta}
& \left\{\calK^w(S;S)\right\}
\ar@`{p+(-16,-16),p+(16,-16)}_{\ast,\#,dm,\ldots}
\ar@/^/[l]^{\delta^{-1}}
\ar@{^{(}->}[r]
& \left\{\calK^w(H;T)\right\},
\ar@`{p+(-16,-16),p+(16,-16)}_{\ast,\#,dm,hm,tm,tha,\ldots}
} \]
via the ``associated graded'' procedure
described in~\cite{WKO2}. Here $\calK^w(S)$ is the set of
$S$-labelled w-tangles~\cite{WKO2}, $\calK^w(H;T)$ is the set of w-knotted
$H$-labelled hoops and $T$-labelled balloons~\cite{KBH}, $\calK^w(S;S)$ is
the same but with $H=T=S$, and $\delta$ is the same as in~\cite{KBH}. This
correspondence is further recalled throughout the rest of this paper.
\Lie The corresponding Lie-theoretic spaces (compare
\cite[Section~\ref{1-subsec:LieAlgebras}]{WKO1}) are
\[ \xymatrix@C=0.8in{
\left\{\calU(I\frakg)^{\otimes S}\right\}
\ar@`{p+(-16,-16),p+(16,-16)}_{\ast,dm,\ldots}
\ar@/^/[r]^-{\delta}
& \left\{\calU(\frakg)^{\otimes S}\otimes\calS(\frakg^\ast)^{\otimes S}\right\}
\ar@`{p+(-16,-16),p+(16,-16)}_{\ast,\#,dm,\ldots}
\ar@/^/[l]^-{\delta^{-1}}
\ar@{^{(}->}[r]
& \left\{\calU(\frakg)^{\otimes H}\otimes\calS(\frakg^\ast)^{\otimes T}\right\}.
\ar@`{p+(-16,-16),p+(16,-16)}_{\ast,\#,dm,hm,tm,tha,\ldots}
} \]
This correspondence is further recalled throughout the rest of this paper.
\AT In~\cite{AT} there is no good counterparts for last two columns of our
diagram. The counterpart of the first (and primary) column is a mixture
$\hat{\calU}((\fraka_n\oplus\attder_n)\ltimes\attr_n)$ containing the most
important spaces occurring in~\cite{AT}. More in the next section.
\subsection{Acknowledgement} This paper was written almost entirely with
Z.~Dancso in the room (physically or virtually via Skype), working on
various parts of our joint series \cite{WKO1}--\cite{WKO3}. Hence her
indirect contribution to it, in a huge number of routine consultations,
should be acknowledged in capitals: THANKS, ZSUZSI. I would like
to further thank A.~Alekseev and S.~Morgan for their comments and
suggestions.