\section{Glossary of notation} \label{sec:glossary}

Icons, then Greek letters, then Latin, and then symbols:

\noindent
{\small\begin{multicols}{2}
\TopLieAT Links with topology, finite-dimensional Lie theory, and the
Alekseev-Torossian paper~\cite{AT}.

\parpic[l]{
  \imagetop{\input{figs/face.pstex_t}}%
  \,\,\imagetop{\input{figs/human.pstex_t}}%
  \,\,\imagetop{\input{figs/machine.pstex_t}}%
  \hspace{-6pt}
}%
\noindent Human input, multi-line human input, and computer output.

\vskip 2mm
\parpic[l]{
  \imagetop{\includegraphics[width=6mm]{figs/NotebookIcon.eps}}%
    \raisebox{-2mm}{\href{\web/AwCalculus.m}{\tiny\tt FL}}%
  \,\,\imagetop{\includegraphics[width=6mm]{figs/NotebookIcon.eps}}%
    \raisebox{-2mm}{\href{\web/AwCalculus.m}{\tiny\tt AC}}%
  \hspace{-6pt}
}%
\noindent Source code quotes from the Mathematica packages
\href{\web/FreeLie.m}{\tt FreeLie.m} and \href{\web/AwCalculus.m}{\tt
AwCalculus.m} \cite{WKO4}.

\raggedright\begin{list}{}{
  \renewcommand{\makelabel}[1]{#1\hfil}
}
\item

% \alpha
\glosi{alpha}{$\alpha$}{a map $\frakt_n\to\calA_{\text{prim}}^w$ / $\calA^u\to\calA^w$}
% \beta

% \gamma
\glosi{TWGamma}{$\Gamma$}{the conversion $\TW_l\to\TW_s$}
\glosi{Gamma}{$\Gamma(\lambda)$}{$\Gamma_1(\lambda)$}
\glosi{Gammat}{$\Gamma_t(\lambda)$}{solution of $e^{-t\partial_\lambda}=C^{\Gamma_t(\lambda)}$}

% \delta
\glosi{Delta}{$\Delta$}{a co-product}
\glosi{delta}{$\delta$}{double all strands $\calA^w(S)\!\to\!\calA^w(S;S)$}

% \epsilon
% \zeta

% \eta
\glosi{eta}{$\eta$}{a co-unit}

% \theta
\glosi{Theta}{$\Theta$}{$\exp(\rightarrowdiagram+\leftarrowdiagram)$}

% \iota
\glosi{ElDef}{$\iota$}{the embedding $\CW\to\calA^w$}

% \kappa

% \lambda
\glosi{expectation}{$\lambda$}{generic element of $\FL(S)^S$}
\glosi{TWLambda}{$\Lambda$}{the conversion $\TW_s\to\TW_l$}
\glosi{Lambda}{$\Lambda(\lambda)$}{$\Lambda_1(\lambda)$}
\glosi{Lambdat}{$\Lambda_t(\lambda)$}{solution of $C^{t\lambda}=e^{-\partial_{\Lambda_t(\lambda)}}$}

% \mu

% \nu
\glosi{nu}{$\nu$}{Kontsevich integral of the unknot}

% \xi
% \omicron

% \pi
\glosi{piA}{$\pi_A$}{projection on ``framing part''}
\glosi{piT}{$\pi_T$}{projection on trees}
\glosi{pidowncap}{$\pi_{\!\downcap}$}{a projection on wheels}

% \rho
\glosi{rho2}{$\rho_2$}{an involution on $\calA^w$}
\glosi{rho3}{$\rho_3$}{a trivolution on $\calA^w(x,y)$}
% \sigma

% \tau
\glosi{tau}{$\tau$}{an involution on $\SolKV$}

% \upsilon

% \phi
\glosi{Phi}{$\Phi$}{a Drinfel'd associator}
\glosi{PhiV}{$\Phi_V$}{an associator in $\calA^w$}

% \chi
% \psi
% \omega
\glosi{expectation}{$\omega$}{generic element of $\CW(S)$}

\item

%a
\glosi{a}{$a,\bar{a},a_i,b,\ldots$}{generic strand labels}
\glosi{uva}{$a$}{the inclusion usual$\hookrightarrow$virtual}
\glosi{A}{$A$}{Abelian lie algebra}
\glosi{fraka}{$\fraka$}{\cite{AT} notation for $A$}
\glosi{adugamma}{$\ad_u^\gamma$}{a derivation on $\FL(T)$}
\glosi{wRels}{$\aAS$}{the directed AS relation}
\glosi{calAw}{$\calA^w$}{arrow-diagram spaces}
\glosi{calAwexp}{$\calA^w_{\exp}$}{exponentials in $\calA^w$}
\glosi{calAwHT}{$\calA^w(H;T)$}{arrow-diagram space on heads-tails skeleton}
\glosi{AwHTexp}{$\calA^w_{\exp}(H;T)$}{exponentials in $\calA^w(H;T)$}

%b
\glosi{B}{$B$}{the ``buckle'' KTG}
\glosi{BCH}{$\BCH$}{the Baker-Campbell-Hausdorff series}
\glosi{BCHtb}{$\BCH_{tb}$}{$\BCH$ relative to $tb$}

%c
\glosi{C}{$C^\lambda$}{conjugating generators by exponentials}
\glosi{Cap}{$\Cap$}{$Z^w$ of a knot-theoretic cap}
\glosi{CP}{$CP$}{the $CP$ relation}
\glosi{Cu}{$C_u^{\gamma}$}{$C^{(u\to\gamma)}$}
\glosi{CW}{$\CW$}{cyclic words}

%d
\glosi{D}{$D$}{a diagram in $\calA^w$}
\glosi{dDelta}{$d\Delta$}{strand doubling in $\calA^w(S)$}
\glosi{ldDelta}{$d\Delta$}{``strand doubling'' in $\TW_l$}
\glosi{deta}{$d\eta$}{strand deletion in $\calA^w(S)$}
\glosi{ldeta}{$d\eta$}{``strand deletion'' in $\TW_l$}
\glosi{dsigma}{$d\sigma$}{strand renaming in $\calA^w(S)$}
\glosi{ldsigma}{$d\sigma$}{``strand renaming'' in $\TW_l$}
\glosi{dA}{$\dA$}{strand adjoint in $\calA^w(S)$}
\glosi{ldA}{$\dA$, $dA^S$}{``strand adjoint'' in $\TW_l$}
\glosi{der}{$\der$}{derivations of $\FL$}
\glosi{atder}{$\atder$}{\cite{AT} notation for $\der$}
\glosi{atdiv}{$\atdiv$}{$\sum_u\atdiv_u$}
\glosi{atdivu}{$\atdiv_u$}{a ``self-action'' map $\FL(S)\to\CW(S)$}
\glosi{dm}{$dm$}{strand stitching in $\calA^w(S)$}
\glosi{dS}{$\dS$}{strand antipode in $\calA^w(S)$}
\glosi{ldS}{$\dS$, $dS^S$}{``strand antipode'' in $\TW_l$}

%e
\glosi{calE}{$\calE$}{the Euler operator}
\glosi{Ef}{$E_f$}{the factored presentation}
\glosi{El}{$E_l$}{the lower-interlaced presentation}
\glosi{Es}{$E_s$}{the split presentation}
\glosi{Eu}{$E_u$}{the upper-interlaced presentation}
\glosi{es}{$e_s$}{a map $\FL(T)^H\to\calA^w_{\exp}(H;T)$}
\glosi{exptn}{$\exp\frakt_n$}{the exponential group of $\frakt_n$}

%f
\glosi{F}{$F$}{solution of the generalized KV equations}
\glosi{fg}{$f,g$}{solution of the original KV equations}
\glosi{FL}{$\FL$}{free Lie algebra}

%g
\glosi{frakg}{$\frakg$}{a finite-dimensional Lie algebra}

%h
\glosi{H}{$H$}{a set of head labels}
\glosi{h}{$h_i$}{head labels}
\glosi{hdeg}{$h^{\deg}$}{degree-scaling}
\glosi{hDelta}{$h\Delta$}{head-strand doubling in $\calA^w(H;T)$}
\glosi{shDelta}{$h\Delta$}{``head-strand doubling'' in $\TW_s$}
\glosi{heta}{$h\eta$}{deleting a head-strand in $\calA^w(H;T)$}
\glosi{sheta}{$h\eta$}{``deleting a head-strand'' in $\TW_s$}
\glosi{hsigma}{$h\sigma$}{head-strand renaming in $\calA^w(H;T)$}
\glosi{shsigma}{$h\sigma$}{``head-strand renaming'' in $\TW_s$}
\glosi{hA}{$hA$}{head-strand adjoint in $\calA^w(H;T)$}
\glosi{shA}{$hA$}{``head-strand adjoint'' in $\TW_s$}
\glosi{hm}{$hm$}{head-strand stitching in $\calA^w(H;T)$}
\glosi{shm}{$hm$}{``head-strand stitching'' in $\TW_s$}
\glosi{hS}{$hS$}{head-strand antipode in $\calA^w(H;T)$}
\glosi{shS}{$hS$}{``head-strand antipode'' in $\TW_s$}

%i
\glosi{Ifrakg}{$I\frakg$}{$\frakg\ltimes\frakg^\ast$}
\glosi{wRels}{$\aIHX$}{the directed IHX relation}

%j
\glosi{j}{$j$}{a ``$\log$-Jacobian'' $\FL\to\CW$}
\glosi{Ju}{$J_u$}{a ``partial Jacobian'' $\FL\to\CW$}
%k

%l
\glosi{ElDef}{$l$}{the lower embedding $\FL(S)^S\to\calA^w$}
\glosi{lie}{$\lie$}{\cite{AT} notation for $\FL$}

%m
%n
%o

%p
\glosi{Aprimw}{$\calA_{\text{prim}}^w$}{the primitives in $\calA^w$}
\glosi{AprimwHT}{$\calA_{\text{prim}}^w(H;T)$}{the primitives in $\calA^w(H;T)$}

%q

%r
\glosi{R}{$R$}{$R(1,2)$}
\glosi{Rab}{$R^{\pm 1}(a,b)$}{$Z^w$ of a single $\pm$ crossing}
\glosi{Rl}{$R_l^\pm$}{$R^{\pm 1}$ in $\TW_l$}
\glosi{Rs}{$R_s^\pm$}{$R^{\pm 1}$ in $\TW_s$}
\glosi{RC}{$RC^{-\lambda}$}{inverse of $C^\lambda$}
\glosi{RCu}{$RC_u^{\gamma}$}{$RC^{(u\to\gamma)}$}

%s
\glosi{S}{$S$}{a set of strands}
\glosi{calS}{$\calS$}{a symmetric algebra}
\glosi{sder}{$\sder$}{``special'' derivations}
\glosi{wRels}{$\aSTU$}{a directed STU relation}

%t
\glosi{T}{$T$}{a set of tail labels}
\glosi{t}{$t_i$}{head labels}
\glosi{tij}{$t_{ij}$}{generators of $t_{ij}$}
\glosi{fraktn}{$\frakt_n$}{the Drinfel'd-Kohno algebra}
\glosi{tDelta}{$t\Delta$}{tail-strand doubling in $\calA^w(H;T)$}
\glosi{stDelta}{$t\Delta$}{``tail-strand doubling'' in $\TW_s$}
\glosi{teta}{$t\eta$}{deleting a tail-strand in $\calA^w(H;T)$}
\glosi{steta}{$t\eta$}{``deleting a tail-strand'' in $\TW_s$}
\glosi{tsigma}{$t\sigma$}{tail-strand renaming in $\calA^w(H;T)$}
\glosi{stsigma}{$t\sigma$}{``tail-strand renaming'' in $\TW_s$}
\glosi{tA}{$tA$}{tail-strand adjoint in $\calA^w(H;T)$}
\glosi{stA}{$tA$}{``tail-strand adjoint'' in $\TW_s$}
\glosi{TAut}{$\TAut$}{the exponential group of $\tder$}
\glosi{tb}{$tb$}{tangential bracket}
\glosi{wRels}{$TC$}{the tails-commute relation}
\glosi{tder}{$\tder$}{tangential derivations}
\glosi{attder}{$\attder$}{\cite{AT} notation for $\tder$}
\glosi{tha}{$tha$}{tail-head action in $\calA^w(H;T)$}
\glosi{stha}{$tha$}{``tail-head action'' in $\TW_s$}
\glosi{thm}{$thm$}{tail-head stitching in $\calA^w(H;T)$}
\glosi{tm}{$tm$}{tail-strand stitching in $\calA^w(H;T)$}
\glosi{stm}{$tm$}{``tail-strand stitching'' in $\TW_s$}
\glosi{tru}{$\tr_u$}{a trace map $\FL(S)\to\CW(S)$}
\glosi{attr}{$\attr$}{\cite{AT} notation for $\CW$}
\glosi{tS}{$tS$}{tail-strand antipode in $\calA^w(H;T)$}
\glosi{stS}{$tS$}{``tail-strand antipode'' in $\TW_s$}
\glosi{expectation}{$\TW$}{trees and wheels}
\glosi{TWl}{$\TW_l$}{domain of $E_l$}
\glosi{Es}{$\TW_s$}{domain of $E_s$}

%u
\glosi{u}{$u$}{the upper embedding $\FL(S)^S\to\calA^w$}
\glosi{unzip}{$u$}{unzip operations}
\glosi{uinT}{$u,v,w$}{tail labels}
\glosi{calU}{$\calU$}{universal enveloping algebra}

%v
\glosi{V}{$V$}{$Z^w$ of a knot-theoretic vertex}
%w

%x
\glosi{xinH}{$x,y,z$}{head labels}

%y

%z
\glosi{ZB}{$Z_B$}{$Z^u$ of the buckle $B$}
\glosi{Zu}{$Z^u$}{the $\calA^u$ counterpart of $Z^w$}
\glosi{Zw}{$Z^w$}{a (universal) $\calA^w_{\exp}$-valued invariant}

\item
\glosi{act}{$\act$}{postfix operator application, ``composition done right''}
\glosi{rightarrowdiagram}{$\rightarrowdiagram$}{a single-arrow diagram}
\glosi{ast}{$\ast$}{the stacking product in $\calA^w(S)$}
\glosi{last}{$\ast$}{the ``stacking product'' in $\TW_l$}
\glosi{jail}{$\#$}{the stacking product in $\calA^w(H;T)$}
\glosi{sjail}{$\#$}{the ``stacking product'' in $\TW_s$}
\glosi{wjail}{$\#$}{a product on $\calA^w(S)$}
\glosi{Box}{$\Box$}{the co-product in $\calA^w(S)$}
\glosi{htBox}{$\Box$}{the co-product in $\calA^w(H;T)$}
\glosi{hdeg}{${-1}^{\deg}$}{degree-scaling with $h=-1$}
\glosi{ob}{$\ob{xy}$}{top-bracket notation}
\glosi{partial}{$\partial$}{the map $\FL(S)^S\to\der_S$}
\glosi{ldeta}{$\remove$}{set minus, array key removal}
\glosi{sqcup}{$\sqcup$}{a disjoint union in $\calA^w(S)$}
\glosi{lsqcup}{$\sqcup$}{``disjoint union'' in $\TW_l$}
\glosi{ssqcup}{$\sqcup$}{``disjoint union'' in $\TW_s$}
\glosi{Brutesqcup}{$\sqcup$}{a union made disjoint}
\glosi{htsqcup}{$\sqcup$}{a disjoint union in $\calA^w(H;T)$}
\glosi{uparrow}{$\uparrow_n$}{a skeleton labelled $S=\{1,\ldots,n\}$}
\glosi{wideparen}{$\wideparen{uvw}$}{a cyclic word}
\glosi{parenl}{$(\lambda;\,\omega)_l$}{generic element in $\TW_l$}
\glosi{parens}{$(\lambda;\,\omega)_s$}{generic element in $\TW_s$}
\glosi{parenu}{$(\lambda;\,\omega)_u$}{element in the domain of $E_u$}
\glosi{tb}{$[\cdot,\cdot]_{tb}$}{tangential bracket}

\item
\glosi{overcrossing}{$\overcrossing$}{an over-crossing}
\glosi{undercrossing}{$\undercrossing$}{an under-crossing}
\glosi{virtualcrossing}{$\virtualcrossing$}{a ``virtual'' crossing}
\glosi{PlusVertex}{$\raisebox{-1mm}{\input{figs/PlusVertex.pstex_t}}$}{the
  knot-theoretic ``vertex''}
\glosi{upcap}{$\upcap$}{a knot-theoretic ``cap''}
\glosi{tetrahedron}{$\tetrahedron$}{unknotted tetrahedron}

\end{list}
\end{multicols}}