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\begin{document}
\newdimen\captionwidth\captionwidth=\hsize
\setcounter{secnumdepth}{4}

\title{Extra Figures}

\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 August 7, 2023, this edition \today.}


\maketitle

\noindent(0)
\[
  \def\goesto{{$\overset{\pi}{\longmapsto} -[[x_1,x_2],[x_2,x_3]]$}}
  \input{figs/ArrowsToLie.pstex_t}
\]

\noindent(1-1) figs/wTFgensWen@:
\[
  \wTF=\CA\!\left\langle\left.\left.
  \raisebox{-6mm}{\input{figs/wTFgensWen_2.pstex_t}}
  \right|
  \parbox{1.9in}{\centering \Rs, R2, R3, R4, OC, CP, FR, $W^2$, CW, TV}
  \right|
  \parbox{0.9in}{\centering $S_e, A_e, u_e, d_e$}
  \right\rangle.
\]

\noindent(1-2)
\begin{center} \input figs/FlipRels2_2.pstex_t \end{center}

\noindent(1-3) figs/CapWen@:
\parpic[r]{$\input{figs/CapWen_2.pstex_t}$}
A cap can slide through a wen, hence a capped wen disappears, as shown
on the right, to be denoted \glost{CW}.

\noindent(1-4) figs/TVRel:
\begin{center} \input figs/TVRel.pstex_t \end{center}

\noindent(1-5) figs/Wen2@:
\begin{center} \input{figs/Wen2_2.pstex_t} \end{center}

\noindent(1-6) figs/SkelGenWen:
\[
  \calS = \CA\!\left\langle\left.
  \raisebox{-2mm}{\input{figs/SkelGenWen.pstex_t}}
  \right|
    \text{$W^2$, CW, TV}
  \right\rangle.
\]

\noindent(1-7)
\begin{center} \input figs/BlueR4.pstex_t \end{center}

\noindent(1-8)
\begin{center} \input figs/BlueR2.pstex_t \end{center}

\noindent(1-9) figs/AdjointV:
\begin{center} \input figs/AdjointV.pstex_t \end{center}

\noindent(1-10) figs/wTFprojgens@ (appears in two places!):

\[ \calA^{(s)w}=\CA\!\left.\left.\left\langle
  \raisebox{-6mm}{\input{figs/wTFprojgens_2.pstex_t}}
  \right|
  \parbox{1.2in}{\centering relations as in Section~\ref{subsubsec:wTFGradRels}}
  \right|
  \parbox{1.2in}{\centering operations as in Section~\ref{subsubsec:wTFGradOps}}
  \right\rangle.
\]

\[ \calA^{(s)w}=\CA\!\left.\left.\left\langle
  \raisebox{-6mm}{\input{figs/wTFprojgens_2.pstex_t}}
  \right|
  \parbox{1.4in}{\centering $\aft$, TC, VI, CP, $W^2$, TW, CW, FR, (RI for $\calA^{sw}$)}
  \right|
  \parbox{1.0in}{\centering $S_e, A_e, u_e, d_e$}
  \right\rangle.
\]

\noindent(1-11) figs/VI@:
\begin{center} \input figs/VI_2.pstex_t \end{center}

\noindent(2-1) figs/ZofWen1:
\begin{center} \input figs/ZofWen1.pstex_t \end{center}

\needspace{20mm}
\noindent(2-2) figs/ZofWen2:
\begin{center} {\def\ol{\overline}\input{figs/ZofWen2.pstex_t}} \end{center}

\noindent(2-3) figs/ZofWen3: (Cancelled)
\begin{center} \input figs/ZofWen3.pstex_t \end{center}

\noindent(2-4) figs/ZofWen4: (Postponed)
%\begin{center} \input figs/ZofWen4.pstex_t \end{center}

\noindent(2-5) figs/ColouredCrossings:
\begin{center} \input{figs/ColouredCrossings.pstex_t} \end{center}

\noindent(3-1) figs/FRCalc:
{
  \def\ol{\overline}
  \def\Rea{$R=e^a$}
  \def\RbarR{$\ol{R}\!=\!e^{-a}\!=\!R^{-1}$}
  \def\BoB{{\raisebox{-1.75mm}{\input{figs/BlueOverBlackMinus.pstex_t}}}}
  \def\ZBoB{$Z\left(\BoB\right)^{21}$}
  \begin{center} \input{figs/FRCalc.pstex_t} \end{center}
  \begin{center} The proof of $(R^{-1})^{21} = Z\left(\BoB\right)$ \end{center}
}

\noindent(3-2) figs/TVUnitarity:
{
  \def\SSV{$S_1S_2(V^{-1})$}
  \def\OV{$\overline{V}$}
  \def\OW{$\overline{W}$}
  \begin{center} \input{figs/TVUnitarity.pstex_t} \end{center}
}

\noindent(3-3) figs/SkelGen:
\[
  \calS^0 = \CA\!\left\langle\raisebox{-2mm}{\input{figs/SkelGen.pstex_t}}\right\rangle
\]

\noindent(4-1) figs/wTFgensWen:
 \[
  \wTF=\CA\!\left\langle\left.\left.
  \raisebox{-2mm}{\input{figs/wTFgensWen.pstex_t}}
  \right|
  \parbox{1.9in}{\centering \Rs, R2, R3, R4, OC, CP, FR, $W^2$, CW, TV}
  \right|
  \parbox{0.7in}{\centering $S_e, u_e, d_e $}
  \right\rangle.
\]

\noindent(4-2) figs/SkelGenWen@:
\[
  \calS = \CA\!\left\langle\left.
  \raisebox{-2mm}{\input{figs/SkelGenWen_2.pstex_t}}
  \right|
    \text{$W^2$, CW, TV}
  \right\rangle.
\]

\noindent(4-3) figs/TheTwistedVertex:
\parpic[l]{\input{figs/TheTwistedVertex.pstex_t}}
The last wen relation describes the interaction of wens and vertices,
as illustrated on the left. In the band notation the non-filled band
represents the larger circle, and the band the inner/smaller circle,
as usual. Conjugating a vertex by three wens switches the top and
bottom bands, as shown in the figure on the left. Alternatively in the
flying circle interpretation, if both rings flip, then merge, and then
the merged ring flips again, this is homotopic to no flips, except the
fly-in direction (from below or from above) has changed. We denote this
relation by \glost{TV}, for {\em Twisted Vertex}. Diagrammatically,
we have obtained the relation shown below: {\color{red} ???}

\noindent(4-4) figs/wTFprojgensWen and figs/wTFprojgens:
\[ \calA^{(s)w}=\CA\!\left.\left.\left\langle
  \raisebox{-2mm}{\input{figs/wTFprojgensWen.pstex_t}}
  \right|
  \parbox{2in}{\centering $\aft$, TC, VI, CP, $W^2$, TW, CW, FR, (RI for $\calA^{sw}$)}
  \right|
  \parbox{0.8in}{\centering $S_e, u_e, d_e$}
  \right\rangle.
\]
\[ \calA^{sw}=\CA\!\left.\left.\left\langle
  \raisebox{-2mm}{\input{figs/wTFprojgens.pstex_t}}
  \right|
  \parbox{1.5in}{\centering $\aft$, TC, VI, CP, RI}
  \right|
  \parbox{0.9in}{\centering $S_e, u_e, d_e, -$}
  \right\rangle.
\]

\noindent(4-5) figs/ZofWen2@:
\begin{center} {\def\ol{\overline}\input{figs/ZofWen2_2.pstex_t}} \end{center}

\needspace{20mm}
\noindent(4-6) figs/TVUnitarity@:
{
  \def\SSV{$S_1S_2(V^{-1})$}
  \def\OV{$\overline{V}$}
  \def\OW{$\overline{W}$}
  \begin{center} \input{figs/TVUnitarity_2.pstex_t} \end{center}
}

\noindent(4-7) figs/NooseEquations, figs/NooseCappedProof, figs/NooseCapped:
\[ \input{figs/NooseEquations.pstex_t} \]
\[ \input{figs/NooseCappedProof.pstex_t} \]
\[ \input{figs/NooseCapped.pstex_t} \]

\noindent(5-1) figs/VertexTypes@:
\[ \input{figs/VertexTypes_2.pstex_t} \]

\noindent(5-2) figs/BandUnzip@:
\[ \input{figs/BandUnzip_2.pstex_t} \]

\noindent(5-3) figs/DetailedVertex@:
\[ \input{figs/DetailedVertex_2.pstex_t} \]

\noindent(5-4) figs/Associator:
\[ \input{figs/Associator.pstex_t} \]

\noindent(5-5) figs/TubeOrientation@:
\[ \input{figs/TubeOrientation_2.pstex_t} \]

\noindent(5-6) figs/BandCrossings@:
\[ \input{figs/BandCrossings_2.pstex_t} \]

\noindent(5-7) figs/PushMembranes@:
\[ \input{figs/PushMembranes_2.pstex_t} \]

\end{document}
\endinput

