\begin{comment}
Let $\mathscr{S}$ be the category whose objects are finite sets and for $X\in \mathscr{S}$, let $FX$ denote the free group on the alphabet $X$. The morphisms on $\mathscr{S}$ are given by  $Hom(X,Y):=Hom(FX,FY)$. Let $G$ be a group. A $G$-computer is a contravariant functor $\Gamma$ from $\mathscr{S}$ to the category of $G$-valued data sets, with $\Gamma : X \mapsto G_X$. Given a free group homomorphism $\mu:FY\to FX$, $\Gamma$ produces a map $\Gamma (\mu):G_X\to G_Y$ in the following way. If $g_1,\dots,g_n$ are elements stored in registers $x_1,\dots,x_n$, and if $\mu(y_i)=w_i(x_1,\dots,x_n)$ is a word in the $x_i$'s, then $\Gamma(\mu)$ inserts $w_i(g_1,\dots,g_n)$ in the register $y_i$.

\begin{displaymath}
\xymatrix{
Y \ar[r]^{\mu} \ar[d]^{\Gamma} & X \ar[d]
\\
G_Y & \ar[l]^{\Gamma(\mu)} G_X
}
\end{displaymath}

The homomorphisms for the operations of Section 2 are given as follows:

$m_z^{xy}:F(\{z\}\cup X) \to F(\{x,y\}\cup X)$, $m_z^{xy}(a)=\begin{cases}
xy& a=z \\
a & a\ne z
\end{cases}$

$S^x:F(\{x\}\cup X) \to F(\{x\}\cup X)$, $S^x(a)=\begin{cases}
a^{-1}& a=x \\
a & a\ne x
\end{cases}$

$e_x:F(\{x\} \cup X) \to FX$, $e_x(a)=\begin{cases}
e & a=x \\
a & a\ne x
\end{cases}$

$\rho_y^x: F(\{y\} \cup X) \to F(\{x\} \cup X)$, $\rho_y^x(a)=\begin{cases}
x & a=y \\
a & a \ne y
\end{cases}$

$d_x: FX \to F(\{x\} \cup X)$, $d_x(a)=a$

$\Delta^x_{yz}: F({\{y,z\}}\cup X) \to F({\{x\}}\cup X)$, $\Delta^x_{yz}(a)=\begin{cases}
x & a=y, a=z \\
a & \text{otherwise}
\end{cases}$

To accommodate the disjoint union map we need to introduce the multi-category of $\mathscr{S}$ (correct wording?), as it is a binary operation.
The homomorphism for the disjoint union $G_X \cup G_Y$ is the canonical isomorphism

$\bigcup_{X,Y} : F(X\cup Y)\to FX\times FY$

A meta-group is a contravariant functor from the multi-category of sets with morphisms the homomorphisms of free groups $m^{xy}_z$, $S^x$, $e_x$, $\rho_y^x$, $d_x$ and $\bigcup$ above, to a category with morphisms induced by those homomorphisms. The axioms omitted in Section 2 come from identities satisfied by those homomorphisms and are forced to hold on \textquoteleft\textquoteleft data set operations" by contravariance. For example, the axiom for the identity element insertion, $e_x \sslash m^{xy}_z=\rho^y_z$ (the statement $eg=g$ for groups) follows from the identity $e_x(m^{xy}_z(z))=e_x(xy)=e_x(x)e_x(y)=ey=y=\rho^y_z(z)$.

\end{comment}


 \begin{figure}
 \centering
 \includegraphics{abelian_monoid.eps}
 \caption{long knots form an abelian monoid under the connected sum operation}
 \end{figure}
The reason $Z^\beta$ makes sense on circular knots is that they are equivalent to long knots, also known as 1-component tangles. Indeed, consider cutting a circular knot at two points. There are two ways one can assemble the resulting two pieces of strand back together, but they are isotopic; one can imagine shrinking the knotted part of one of the strands to a very tiny size, sliding it all along the other strand, and growing it back on the other side. Put differently, long knots form an abelian monoid under the connected sum operation.

The analogue for links is not as immediate but nevertheless straightforward. The issue is that there are several inequivalent ways to cut an $n$-component link in $n$ places to obtain an $n$-component tangle. But nothing prevents us from taking a quotient of the space of $n$-component tangles by the image of all the maps from $n+1$-component tangles which connect two components together; then in the quotient all ways of cutting a link open are equivalent. The good news is that the procedure is easily paralleled on the $\beta$ calculus, since connecting two strands simply means multiplying two labels. Also, the map $Z^\beta$ clearly descends to the quotients.


\begin{comment}
One expects in light of theorem 1 that an extension of $Z^\beta$ to links contains the multivariable Alexander polynomial. While no formal proof has been written, a computer experiment has shown without shadow of doubt that this is the case. The interesting question is how much more information is contained in $Z^\beta$, if any at all.
\end{comment}

\begin{comment} Unfortunately, there is no simple analog relating $n$-component links to $n$-component tangles and as a consequence $Z^\beta$ is not well-defined on links. Nevertheless, since the classical Alexander polynomial admits a multivariable generalization to links, one expects in light of Theorem 1 that at least some invariant information can be obtained from $Z^\beta$ on links. A computer experiment shows without shadow of doubt that $Z^\beta$ contains the multivariable Alexander polynomial itself, but a proof remains to be found.
\end{comment}

\begin{comment}
Armed with this new word in our vocabulary we can now define what seems to be the \textquoteleft\textquoteleft most natural\textquoteright\textquoteright \space meta-group: the meta-group of oriented coloured v-tangles. Let $\Gamma_X$ be the set of v-tangles with strands labelled by $X$. There is a natural definition for all the meta-group operations. The multiplication $m_z^{xy}$ \footnotemark \space concatenates strand $x$ with strand $y$ and labels the resulting strand $z$ (note that we \textit{need} virtual tangles for this to be well-defined), $S^x$ reverses the orientation of strand $x$, $e^x$ creates an isolated strand with label $x$, $d^x$ deletes strand $x$, and $\Delta_{yz}^x$ is the cabling operation with input strand $x$ and output strands $y$ and $z$. [EXPAND]
\footnotetext{Remark: this is \textit{not} a meta-generalization of the group structure on braids.}
\end{comment}

As mentioned and made explicit above, the $\beta$ calculus is equipped with ugly operations. However, it can be implemented in a computer program in a very short paragraph, and the program handles the proof of Theorem 1 (and presumably also of Conjecture 1) very well. The \textit{Mathematica} code showcased in Figure \ref{fig:program} produces a ready-to-use program with neatly formatted output.
%\begin{figure}[h]
%\includegraphics[scale=0.5]{beta_demo_initialization.eps}
%\includegraphics[scale=0.5]{beta_demo_program.eps}
%\caption{The computer program}\label{fig:program}
%\end{figure}

Let us illustrate the method with the important swap map axiom $tm^{12}_1 \sslash sw_{14}=sw_{14} \sslash sw_{24} \sslash tm^{12}_1$.  We can use the computer program to check it on an arbitrary $3\times 2$ array (i.e. an array with one more than the number of \textquoteleft\textquoteleft participating" indices of each type), as in Figure \ref{fig:check}
\begin{figure}[h]
\includegraphics[scale=0.5]{sw1_check.pdf}
\caption{Checking relation 4a}\label{fig:check}
\end{figure}

We claim that this constitutes a proof that the identity holds on arrays of arbitrary dimension. The key lies in the fact that the operations are linear in the \textquoteleft \textquoteleft non-participating" indices. It is very clear then, from the 2-variable polynomial point of view, that the result still holds if one replaces a non-participating entry by an arbitrary sum. The argument applies to the other axioms as well and the reader is welcome to verify them.

 \begin{comment}
(2) (SKETCH) The key is to show that a single crossing gets mapped to (essentially) its Burau representation via $Z^\beta$ and that concatenating braids (essentially) corresponds to matrix multiplication of the \textquoteleft\textquoteleft matrix part\textquoteright\textquoteright \space of $Z^\beta$.

(3) (WISHFUL) Note first that a priori $Z^\beta$ does not make sense on round knots, as one would have to multiply all the labels together (including the last remaining one with itself). It turns out however that round knots are equivalent to long knots, so one can simply pick an arbitrary\footnotemark \footnotetext{See section \ref{ssec:links}} point to cut the knot open and compute $Z^\beta$ without ambiguity. However to prove the assertion, given a knot $K$, turn it first into a braid $b_K$ via Alexander's theorem. Compute $Z^\beta(b_K)$ to obtain the Burau representation, as in (2). Then multiply all the strands together except the last one; this operation is equivalent to taking the determinant of an $(n-1) \times (n-1)$ minor.
\end{comment}
%Hard-coding: relations 4a & 4b of Figure 4
%note extensive presence of large chunks of commented material with \begin{comment} .... \end{comment}


%reference required for discussion of v-tangles
%paragraph on computational advantages following the section on Z^\beta should probably be rewritten

%The bicrossed product construction is easily incorporated in this scheme. Recall that in a bicrossed product $(H\times T)_{sw}$, elements have unique representations of the form $th$ and $h^\prime t^\prime$ where $t,t^\prime \in T$, $h, h^\prime \in H$.

%We will call a meta-group equipped with an additional map $\Delta_{yz}^x$ satisfying the relations of the $\Delta_{yz}^x$ homomorphism a Hopf meta-group.

%\footnotetext{To avoid cluttering notation, we write $\{x,y\}$ for a set containing $x$ and $y$, and possibly more elements. We also assume common sense with respect to naming: for instance, if an operation creates the register $x$, we assume it did not exist before.}