\begin{frame}
{\LARGE 3. An Implementation of $\calA$}
If I didn't implement I wouldn't believe myself.
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Written in Mathematica~\cite{Wolfram:Mathematica}, available as the notebook {\tt Alpha.nb} at \url{http://drorbn.net/mo21/ap}. Code lines are highlighted in grey, demo lines are plain.
We start with an implementation of elements (``{\tt Wedge}'') of exterior algebras, and of the wedge product (``{\tt WP}''):
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\begin{frame}\null
We then define the exponentiation map in exterior algebras (``{\tt WExp}'') by summing the series and stopping the sum once the current term (``{\tt t}'') vanishes:

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Contractions!

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\begin{frame}
\parpic[r]{\def\X{$X_{ijkl}[S,T]$} \input{figs/Xp.pdf_t}}
$\calA${\tt [is,os,cs,w]} is also a container for the values of the $\calA$-invariant of a tangle. In it, {\tt is} are the labels of the input strands, {\tt os} are the labels of the output strands, {\tt cs} is an assignment of colours (namely, variables) to all the ends $\{\xi_i\}_{i\in\text{\tt is}}\sqcup\{x_j\}_{j\in\text{\tt os}}$, and {\tt w} is the ``payload'': an element of $\Lambda\left(\{\xi_i\}_{i\in\text{\tt is}}\sqcup\{x_j\}_{j\in\text{\tt os}}\right)$.

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\begin{frame}\null
The negative crossing and the ``point'':
\[ \def\Xbar{$\bar{X}_{ijkl}[S,T]$}  \def\P{$P_{ij}[T]$}
  \input{figs/XmP.pdf_t}
\]

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The linear structure on $\calA$'s:

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Deciding if two $\calA$'s are equal:

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\begin{frame}\null
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\null\hfill\smash{\imagetop{\input{figs/R2Left.pdf_t}}}
\newline The union operation on $\calA$'s (implemented as ``multiplication''):

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Contractions of $\calA$-objects:

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Automatic and intelligent multiple contractions:

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\begin{frame}{\LARGE 4. Skein relations and evaluations for $\calA$}
\[ \input{figs/SimpleTangle.pdf_t} \]

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\begin{frame}{\large Reidemeister 2}

\[ \input{figs/R2.pdf_t} \]

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\end{frame}
\begin{frame}{\large Reidemeister 3}

\[ \input{figs/R3.pdf_t} \]

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\end{frame}

\begin{frame}{\large Reidemeister 1}
\[ \def\p{$=\tau_1^{1/2}$} \def\m{$=\tau_1^{-1/2}$} \input{figs/R1.pdf_t} \]

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(So we have an invariant, up to rotation numbers).
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\begin{frame}{\large The Relation with the Multivariable Alexander Polynomial}
\[ \input{figs/Borromean.pdf_t} \]

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\begin{frame}{\large Overcrossings Commute but Undercrossings don't}

\[ \input{figs/OUC.pdf_t} \]

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\begin{frame}{\large The Conway Relation\hfill (see~\cite{Conway:Enumeration})}
\[ \input{figs/Conway.pdf_t} \]

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\[ \includegraphics[height=36mm]{../../Projects/Gallery/Conway.png} \]
\end{frame}

\begin{frame}{\large Conway's Second Set of Identities\hfill (see~\cite{Conway:Enumeration})}
\[ \def\b{$=((uv)^{1/2}+(uv)^{-1/2})$} \def\c{$=((u/v)^{1/2}+(u/v)^{-1/2})$} \input{figs/Conway2nd.pdf_t} \]

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\begin{frame}\null
{\bf Virtual versions} (Archibald,~\cite{Archibald:Thesis})
\[ \def\b{$=(\tau_1^{1/2}+\tau_1^{-1/2})$} \def\c{$=(\tau_2^{1/2}+\tau_2^{-1/2})$} \input{figs/Conway2ndV.pdf_t} \]

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\end{frame}

\begin{frame}{\large Conway's Third Identity \hfill (see~\cite{Conway:Enumeration})}
\[ \input{figs/C3.pdf_t} \]

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\begin{frame}\null
{\bf Virtual version} (Archibald,~\cite{Archibald:Thesis})
\[ \input{figs/C3V.pdf_t} \]

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\end{frame}

\begin{frame}{\large Jun Murakami's Fifth Axiom \hfill (see~\cite{MurakamiJ:StateModel})}
\[ \def\prop{$=\frac{\sqrt{S}(1-T)}{\sqrt{T}}$} \input{figs/MJ5.pdf_t} \]

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\[ \includegraphics[height=20mm]{../../Projects/Gallery/MurakamiJ.jpg} \]
\end{frame}

\begin{frame}\null
{\bf Virtual versions} (Archibald,~\cite{Archibald:Thesis})
\[ \def\prop{$=(T^{-1/2}-T^{1/2})$} \input{figs/MJ5V.pdf_t} \]

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\end{frame}

\begin{frame} {\large Jun Murakami's Third Axiom\hfill(see~\cite{MurakamiJ:StateModel})}
\[ \scalebox{0.64}{\input{figs/Murakami3.pdf_t}} \]

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\end{frame}

\begin{frame}{\large The Naik-Stanford Double Delta Move\hfill(see~\cite{NaikStanford:Move})}
\vskip -5mm
\[
  \includegraphics[height=20mm]{../../Projects/Gallery/Naik.jpg}
  \qquad\scalebox{0.8}{\input{figs/NaikStanford.pdf_t}}\qquad
  \includegraphics[height=20mm]{../../Projects/Gallery/Stanford.jpg}
\]

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\end{frame}

\begin{frame}\null
{\bf Virtual Version 1} (Archibald,~\cite{Archibald:Thesis})

\[ \input{figs/VNS1.pdf_t} \]

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\begin{frame}\null
{\bf Virtual Version 2} (Archibald,~\cite{Archibald:Thesis})

\[ \input{figs/VNS2.pdf_t} \]

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\end{frame}