\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.