spinnetsintro.tex
The tools used in this paper to explore $\SL$-character varieties are \emph{trace diagrams}, which are a slight generalization of \emph{spin networks}, which will be formally defined in Definitions \ref{def:spinnet} and \ref{def:tracediagram}.

Informally, a spin network is a graph that can also be interpreted as a function. Depending upon the type of diagram, the function may be between tensor products of $V$, or between tensor powers of $\SL$ representations, which are symmetric powers of $V$. In order for a spin network's function to be well-defined, the inputs and outputs of the function must be specified, and the graph must be given a small amount of extra structure. Once this is done, algebraic rules become diagrammatic rules that are much easier to work with.

Trace diagrams generalize spin networks by allowing matrices to act on specific strands of the diagram. For a specific choice of matrices, the diagrams are still functions as above. However, when the diagram has no inputs or outputs, it becomes a function $\SL\times\cdots\times\SL\to\C$ that is invariant under simultaneous conjugation.

%Elements of $\SL$ are drawn as circled variables along the strands, while elements of the strand space $V$ or $\mathrm{Sym}^n(V)$ will be drawn in boxes at the end of the strands.

\begin{figure}[htb]
$$\tikz[trivalent,shift={(0,.5)}]{ \draw(-.5,-1)node[leftlabel]{1}to[bend left](0,-.5); \draw(.5,-1)node[rightlabel]{3}to[bend right](0,-.5); \draw(0,-.5)to[wavyup]node[leftlabel]{3}(0,0)to[bend left]node[matrix]{g}(-1,1)node[leftlabel]{2}; \draw(0,0)to[bend right]node[rightlabel]{5}(.5,.5)to[bend right](1,1)node[rightlabel]{3}; \draw(.5,.5)to[bend left](0,1)node[leftlabel]{4}; } \qquad\qquad \tikz[trivalent,every node/.style={basiclabel},scale=1.4]{ \draw(0,.5)circle(.4)(0,.1)arc(-145:145:.7); \node[matrix]at(-.4,.5){g}; \node[matrix]at(.4,.5){h}; \node at(-.4,1){2};\node at(.5,.95){3};\node at(1.2,1.1){4}; }$$
\caption{Example of a spin network representing a function from $V_1\otimes V_3$ to $V_2\otimes V_3\otimes V_4$, and a trace diagram representing a function $\SL\times\SL\to\C$.}\label{f:spinnet-tracediagram-example}
\end{figure}

Diagrammatic notations such as spin networks were developed by physicists in the middle of the 20th century as a graphical description of quantized angular momentum. The theory has been developed extensively since that time, most notably by Cvitanovic \cite{Cv08} and Stedman \cite{ste90}. There are several names in the literature for the particular case of trace diagrams. They are also sometimes referred to as birdtracks'' or simply tensor diagrams'' \cite{Cv08}. Trace diagram calculations are very similar to skein module calculations in knot theory. Indeed, the Kauffman bracket skein module can be thought of as a quantization of the diagrams in this paper \cite{Kau91}.

This section follows the diagrammatic conventions of \cite{LP,Pet06,Pet09}, rather than the more common conventions of \cite{CFS95,Kau91,Ma99}. The most obvious difference is \eqref{eq:capcup}, which means that several diagrams will differ in sign from these sources. Proofs are omitted, as they do not contribute to the main point of the paper. We refer the reader to \cite{LP,Pet06,Pet09} for these. Similar proofs for alternate diagrammatic conventions are in \cite{CFS95,Kau91}.

\subsection{Diagrammatic Description of $V_n$}\label{ss:diagrammatic-symmetrization}

Diagrammatically, vectors $v\in V$ are represented by $\tikz{\draw(0,.2)node[vector]{$v$}to(0,.8);}$, and dual vectors $v^*\in V^*$ by $\tikz{\draw(0,.2)to(0,.8)node[vector]{$v$};}$. Strands without vector labels are unknown vectors. Tensor products are represented by drawing diagrams adjacent to each other:
$v_1\otimes v_2\equiv\tikz{\draw(0,.2)node[small vector]{$v_1$}to(0,.8)(.5,.2)node[small vector]{$v_2$}to(.5,.8);}$.
The symmetrization operation is indicated by
\begin{equation}\label{eq:symmetrizer-diagram}
\tikz[xscale=.7]{
\foreach\xa in{1,1.3,2.7,3}{\draw(\xa,0)to(\xa,1);}
\draw[draw=black,fill=gray!10](.8,.35)rectangle(3.2,.75);\node[basiclabel]at(2,.55){$n$};
\node[basiclabel]at(2,.1){$\cdots$};
}
=\frac{1}{n!}\sum_{\sigma\in\Sigma_n}
\tikz[xscale=.7]{
\foreach\xa in{1,1.3,2.7,3}{\draw(\xa,0)to(\xa,1);}
\draw[draw=black,fill=gray!10](.8,.35)rectangle(3.2,.75);\node[basiclabel]at(2,.55){$\sigma$};
\node[basiclabel]at(2,.1){$\cdots$};
},
\end{equation}
which is an \emph{idempotent} since
$$\tikz[xscale=.7]{ \foreach\xa in{1,1.3,2.7,3}{\draw(\xa,0)to(\xa,1);} \draw[draw=black,fill=gray!10](.8,.35)rectangle(3.2,.75);\node[basiclabel]at(2,.55){n}; \node[basiclabel]at(2,.1){\cdots}; }\circ\tikz[xscale=.7]{ \foreach\xa in{1,1.3,2.7,3}{\draw(\xa,0)to(\xa,1);} \draw[draw=black,fill=gray!10](.8,.35)rectangle(3.2,.75);\node[basiclabel]at(2,.55){n}; \node[basiclabel]at(2,.1){\cdots}; }=\tikz[xscale=.7]{ \foreach\xa in{1,1.3,2.7,3}{\draw(\xa,0)to(\xa,1);} \draw[draw=black,fill=gray!10](.8,.35)rectangle(3.2,.75);\node[basiclabel]at(2,.55){n}; \node[basiclabel]at(2,.1){\cdots}; }.$$

The basis elements for $V_n$ and $V_n^*$ described in section \ref{ss:reptheory} take the form
\begin{equation}\label{eq:symmetric-basis}
{\sf n}_{n-k}=
\tikz[xscale=.7]{
\foreach\xa/\xb in{1/$e_1$,3/$e_1$,4/$e_2$,6/$e_2$}{
\draw(\xa,0)node[small vector]{\xb}to(\xa,1);
}
\draw[draw=black,fill=gray!10](.8,.45)rectangle(6.2,.8);\node[basiclabel]at(3.5,.625){$n$};
\node[basiclabel]at(2,0){$\cdots$};\node[basiclabel]at(5,0){$\cdots$};
\draw[decorate,decoration=brace](3.2,-.35)to node[auto,basiclabel]{$n-k$}(.8,-.35);
\draw[decorate,decoration=brace](6.2,-.35)to node[auto,basiclabel]{$k$}(3.8,-.35);
}
{\sf n}^*_{n-k}=
\tikz[xscale=.7,yscale=-1,shift={(0,-.5)}]{
\foreach\xa/\xb in{1/$e_1$,3/$e_1$,4/$e_2$,6/$e_2$}{
\draw(\xa,0)node[small vector]{\xb}to(\xa,1);
}
\draw[draw=black,fill=gray!10](.8,.45)rectangle(6.2,.8);\node[basiclabel]at(3.5,.625){$n$};
\node[basiclabel]at(2,0){$\cdots$};\node[basiclabel]at(5,0){$\cdots$};
\draw[decorate,decoration=brace](.8,-.35)to node[auto,basiclabel]{$n-k$}(3.2,-.35);
\draw[decorate,decoration=brace](3.8,-.35)to node[auto,basiclabel]{$k$}(6.2,-.35);
}.
\end{equation}

It is also customary to represent elements of $V_n$ and $V_n^*$ by thicker strands labeled by $n$, so that the basis elements just described may also be written
\begin{equation}\label{eq:thick-strands}
{\sf n}_i = \tikz[trivalent]{\draw(0,0)node[small vector]{${\sf n}_i$}to(0,1)node[rightlabel]{$n$};}
{\sf n}^*_i = \tikz[trivalent]{\draw(0,0)node[rightlabel]{$n$}to(0,1)node[small vector]{${\sf n}_i$};}
\end{equation}

The pairing between $V_n$ and $V_n^*$ is given by
$${\sf n}^*_{n-k}({\sf n}_{n-l}) =\tikz[trivalent]{\draw(0,-.2)node[small vector]{{\sf n}_{n-l}}to(0,1.2) node[rightlabel,pos=.6,scale=.8]{n}node[small vector]{{\sf n}_{n-k}};} =\raisebox{2pt}{\delta_{kl}}\!\Big/\!\raisebox{-2pt}{\tbinom{n}{k}}.$$
Expanding the symmetrizers shows visually the reason for the term $\tbinom{n}{k}$.

The $\SL$-action on $V_n$ is represented by a matrix $g\in\SL$ drawn on the strand:
$$g\cdot {\sf n}_{n-k} = \tikz[trivalent]{\draw(0,-.2)node[small vector]{{\sf n}_{n-k}} to(0,.6)node[matrix]{g}to(0,1.2)node[rightlabel]{n};}$$

\subsection{Spin Networks}
The formal definitions of spin networks and trace diagrams follow.

\begin{definition}\label{def:spinnet}
A \emph{spin network} is a planar trivalent graph with edges labeled by representations of $\mathrm{SL}(2,\mathbb{C})$, such that the labels adjoining each vertex form an admissible triple. The graph is drawn with all free ends at the top or the bottom, and all local extrema are in general position relative to the orientation of the diagram.
\end{definition}

\begin{definition}\label{def:tracediagram}
A \emph{trace diagram} is a spin network with one or more edges marked by elements of the matrix group $\mathrm{SL}(2,\mathbb{C})$, with no marking occurring at a local extrema.
\end{definition}

Every spin network and trace diagram corresponds to a unique function. The domain of the function is a tensor product of irreducible $\SL$-representations, given by the labeling of the input edges, while the co-domain is given by the labeling of the output edges.

The function may be computed by decomposing the diagram into horizontal slices. The diagram's function is then the composition of the functions of the individual slices. (One can also decompose the diagram into vertical slices, which are conjoined as tensor products.)

We now describe the lexicon for reading off a spin network's function. First, the single-strand \emph{cup} and \emph{cap} diagrams are
\begin{equation}\label{eq:capcup}
\tikz{\draw(0,.7)arc(180:360:.5);} = e_1\otimes e_2-e_2\otimes e_1
\tikz{\draw(0,.2)arc(180:0:.5);} : v\otimes w \mapsto \det[v\: w],
\end{equation}
respectively. Building upon this definition, the symmetrized cup and cap maps are
\begin{equation}\label{eq:trivalent-cap}
\tikz[trivalent]{\draw(0,.7)arc(180:360:.5)node[rightlabel]{$n$};}\equiv
\tikz{\draw(-.1,1)--(-.1,.8)arc(180:360:.3)--(.5,1);\draw(-.7,1)--(-.7,.8)arc(180:360:.9)--(1.1,1);
\draw[draw=black,fill=gray!10](.05,.7)rectangle(.35,-.2);\node[basiclabel,scale=.9]at(.2,.25){$n$};
\node[basiclabel,scale=.7]at(-.375,.8){$\cdots$};\node[basiclabel,scale=.7]at(.825,.8){$\cdots$};}
\tikz[trivalent]{\draw(0,.2)arc(180:0:.5)node[rightlabel]{$n$};}\equiv
\tikz[yscale=-1,shift={(0,-.8)}]{\draw(-.1,1)--(-.1,.8)arc(180:360:.3)--(.5,1);\draw(-.7,1)--(-.7,.8)arc(180:360:.9)--(1.1,1);
\draw[draw=black,fill=gray!10](.05,.7)rectangle(.35,-.2);\node[basiclabel,scale=.9]at(.2,.25){$n$};
\node[basiclabel,scale=.7]at(-.375,.8){$\cdots$};\node[basiclabel,scale=.7]at(.825,.8){$\cdots$};}
.
\end{equation}

From \eqref{eq:capcup} and the normal action of $\SL$ on $V=\C^2$, one can deduce that for $g\in\SL$,
\begin{equation}\label{eq:trace-diagram}
\tikz{\draw(0,.5)circle(.5);\draw(.5,.5)node[matrix]{$g$};} = \tr(g).
\end{equation}
This is why the diagrams are referred to as \emph{trace diagrams}.

Using Schur's Lemma, the Clebsch-Gordan decomposition \eqref{eq:clebsch-gordan} implies that for each admissible triple $\{a,b,c\}$, as specified in Definition \ref{def:admissible}, there are unique (up to a nonzero multiple) injections $V_c\hookrightarrow V_a\otimes V_b$ and projections $V_a\otimes V_b\twoheadrightarrow V_c$. These are sometimes called \emph{intertwining operators}. The injection has the following diagrammatic form:
\begin{equation}\label{eq:trivalent-vertex}
\tikz[trivalent,shift={(0,.5)}]{
\foreach\xa/\xb in{150/a,30/b,-90/c}{\draw[rotate=\xa](0,0)--(0:.8)(0:1)node[basiclabel]{$\xb$};}}
=
\tikz[scale=.6,shift={(0,1)}]{
\foreach\xa in{0,120,240}{
\draw[rotate=\xa](28:2.2)to[bend right](-88:2.2)(0:2.5)to[bend right](-60:2.5);}
\foreach\xa/\xb in{30/b,150/a,270/c}{
\draw[rotate=\xa,draw=black,fill=gray!10](1.2,1.5)rectangle(1.8,-1.5);
\draw[rotate=\xa](0:1.52)node[basiclabel]{$\xb$};
\draw[rotate=\xa](60:1.25)node[basiclabel,rotate=\xa,rotate=60]{$\cdots$};
}
}
\end{equation}
The projection $V_a\otimes V_b\twoheadrightarrow V_c$ is obtained by vertical reflection.

\begin{remark}
The geometric meaning of admissibility is shown in Figure \ref{f:trivalentvertex}. If each edge is expanded into the number of strands specified by its label, then a triple is admissible if and only if there is a way to connect the three sets of strands.
\begin{figure}[htb]
$$\tikz[shift={(0,.5)}]{ \foreach\xa/\xb/\xc in{-100/160/red,150/35/blue,140/45/blue,25/-90/brown,15/-80/brown}{ \draw[color=\xc](\xa:.8)to[bend right](\xb:.8); } \node[basiclabel,rotate=60]at(150:1.2){\overset{a=3}{\overbrace{}}}; \node[basiclabel,rotate=-60]at(30:1.2){\overset{b=4}{\overbrace{}}}; \node[basiclabel]at(-90:1.2){\underset{c=3}{\underbrace{}}}; \draw[blue](90:.8)--(90:1.5)node[basiclabel,anchor=south]{\mathfrak{e}_c(a,b)=2}; \draw[red](-150:.8)--(-150:1.5)node[leftlabel]{\mathfrak{e}_b(a,c)=1}; \draw[brown](-30:.8)--(-30:1.5)node[rightlabel]{\mathfrak{e}_a(b,c)=2}; }$$
\caption{Admissibility implies there is a way to connect sets of $a$, $b$, and $c$ strands as shown.}\label{f:trivalentvertex}
\end{figure}
\end{remark}

This motivates the following notation, which will be used extensively in later sections:
\begin{notation}
Given an admissible triple $\{a,b,c\}$, define
$$\mathfrak{e}_a(b,c) = \frac{b+c-a}{2}, \qquad \mathfrak{e}(a,b,c) = \frac{a+b+c}{2}.$$
\end{notation}
In the figure $\mathfrak{e}_a(b,c)$ represents the (unique) number of strands connecting the $b$ and $c$ edges in the expansion. In these terms, the admissibility condition in Definition \ref{def:admissible} can be reformulated as
$$\mathfrak{e}_a(b,c) \geq 0, \quad \mathfrak{e}_b(a,c) \geq 0, \quad \mathfrak{e}_c(a,b) \geq 0.$$

\subsection{Spin Network Relations}

These basic maps allow the function of a trace diagram or spin network to be written out explicitly. What makes the diagrammatic language powerful, however, is not the ability to represent functions, but rather the ability to manipulate functions in the diagrammatic language.

Indeed, because the symmetrizer is an idempotent, the underlying definitions \eqref{eq:trivalent-cap} and \eqref{eq:trivalent-vertex} are usually unnecessary, as all translation between the algebraic domain and the

Spin networks satisfy certain \emph{skein relations} that can be leveraged to reason about the underlying functions. More precisely, we define the \emph{spin network skein module} to be the space of formal sums of spin networks with coefficients in $\mathbb{C}$, modulo certain relations arising from the representation theoretic understanding.

The relations that follow can all be proven directly from spin network definitions. Proofs are contained in \cite{LP} and earlier works.

A general spin network skein relation has the form $\sum_m \alpha_m \mathsf{s}_m = 0$, where $\alpha_m\in\mathbb{C}$ and each $\mathsf{s}_m$ represents a diagram. Such relations are well-behaved under reflections:

\begin{proposition}[Spin Network Reflection, Proposition 3.6 in \cite{Pet06}]\label{p:spinnetreflection}
Given a spin network $\mathsf{s}$, denote by $\mathsf{s}^\updownarrow$ the spin network obtained by vertical reflection of $\mathsf{s}$, and by $\mathsf{s}^\leftrightarrow$ the spin network obtained by horizontal reflection of $\mathsf{s}$. Then
$$\sum_m \alpha_m \mathsf{s}_m = 0 \quad \Leftrightarrow \quad \sum_m \alpha_m \mathsf{s}^\updownarrow_m = 0 \quad \Leftrightarrow \quad \sum_m \alpha_m \mathsf{s}^\leftrightarrow_m = 0.$$
\end{proposition}
\begin{remark}
In \cite{Pet06}, the third relation is given as $\sum_m (-1)^{|\mathsf{s}_m^v|}\alpha_m\mathsf{s}_m^\leftrightarrow$, where $|\mathsf{s}_m^v|$ represents the number of local extrema in the diagram $\mathsf{s}_m$. However, given any diagrammatic relation, the number of inputs and outputs on each $\mathsf{s}_m$ must be the same. Therefore, the number of local extrema in each case must be either odd or even for all diagrams with this number of inputs and outputs. This means the term can be factored out.
\end{remark}

Other kinds of topological moves may introduce signs. This is the case for local extrema introduced into a strand, crossings of edges adjacent to a vertex, and reorientation of vertices, as indicated in Figure \ref{f:signs}.
\begin{figure}[htp]
$$\tikz{\draw(0,0)to[wavyup](.1,1);} = -\tikz{\draw(.1,0)to[out=90,in=-110](0,.5)to[out=70,in=-110,looseness=2](.4,.5)to[out=70,in=-90,looseness=1](.3,1);} \qquad \tikz[xscale=.6]{\draw(0,.2)arc(180:0:.5);} = -\tikz[shift={(0,.4)}]{\coordinate(vxa)at(0,.55)edge[bend right=45](-.3,.2)edge[bend left=45](.3,.2); \draw(-.3,-.4)to[wavyup](.3,.2)(.3,-.4)to[wavyup](-.3,.2);}$$
\caption{Diagrammatic moves on single strands in a spin network that introduce signs.} \label{f:signs}
\end{figure}
The precise statement and proof of the sign changes follows. We define $\mfs_a(b,c) \equiv (-1)^{\mfe_a(b,c)}$.

\begin{proposition}[Spin Network Sign Changes, Proposition 3.22 in \cite{LP}]\label{p:spinnetsigns}
\begin{align}
\tikz[trivalent]{
\draw(.1,0)to[out=90,in=-110](0,.5)to[out=70,in=-110,looseness=2](.4,.5)to[out=70,in=-90,looseness=1](.3,1)node[rightlabel]{$n$};}
\label{eq:kink}
& = (-1)^n \tikz[trivalent]{\draw(0,0)to[wavyup](.1,1)node[rightlabel]{$n$};}
\\
\tikz[trivalent]{
\coordinate(vxa)at(0,.55)
edge[bend right]node[leftlabel,pos=1]{$a$}(-.4,0)edge[bend left]node[rightlabel,pos=1]{$b$}(.4,0)
edge[]node[rightlabel,pos=1]{$c$}(0,1);}
\label{eq:crossedvertex}
& = \mfs_c(a,b)
\tikz[trivalent,shift={(0,.2)}]{
\coordinate(vxa)at(0,.55)
edge[bend right](-.3,.2)edge[bend left](.3,.2)edge[]node[rightlabel,pos=1]{$c$}(0,1);
\draw(-.3,-.4)node[leftlabel]{$a$}to[wavyup](.3,.2)(.3,-.4)node[rightlabel]{$b$}to[wavyup](-.3,.2);}
= \mfs_a(b,c)
\tikz[trivalent]{
\coordinate(vxa)at(0,.45)
edge[]node[leftlabel,pos=1]{$a$}(0,0)
edge[bend left](-.2,.7)edge[bend right](.2,.65);
\draw(-.2,.7)to[wavyup](-.1,1.1)node[leftlabel]{$c$}
(.45,0)node[rightlabel]{$b$}to[wavyup](.55,.6)to[bend right=90,looseness=2](.2,.65);}
\end{align}
\end{proposition}
A more general version of this result follows. See Figure \ref{f:signs} for the meaning of kink' and crossed extrema'.
\begin{proposition}\label{p:spinnetsignstrong}
Let $\mathsf{T}_1$ and $\mathsf{T}_2$ be topologically equivalent trace diagrams. Then
$$\mathsf{T}_1=(-1)^{k+x}\mathsf{T}_2,$$
where $k$ is the number of kinks'' and $x$ the number of crossed extrema'' in the diagrams obtained by expanding edges as in Figure \ref{f:trivalentvertex}.
\end{proposition}

The following two propositions describe the most basic diagrams, as well as how to join and separate strands in spin networks.

\begin{proposition}[Closed Spin Networks, Proposition 3.19 in \cite{LP}]\label{p:deltatheta}
\begin{align*}
\tikz[trivalent]{\draw(0,.5)circle(.5);\node[basiclabel]at(.5,1){$c$};}
& \equiv \Delta(c) = c+1 = \mathsf{dim}(V_c); \\
\tikz[trivalent,every node/.style={basiclabel}]{
\draw(0,.5)circle(.4)(0,.1)arc(-145:145:.7);
\node at(-.5,.85){$a$};\node at(.5,.85){$b$};\node at(1.2,1.1){$c$};}
& \equiv \Theta(a,b,c) = \frac{\edges abc! \edges bac! \edges cab! (\mfe(a,b,c)+1)!}{a!b!c!}.
\end{align*}
\end{proposition}

\begin{proposition}[Bubble/Fusion Identities, Propositions 3.20,3.21 in \cite{LP}]\label{p:bubblefusion}
\begin{align}
\tikz[trivalent,every node/.style={basiclabel}]{
\draw(0,.5)circle(.3);\node at(-.5,.7){$a$};\node at(.5,.7){$b$};
\draw(-.05,-.2)node[anchor=west]{$c$}to[wavyup](0,.2)(0,.8)to[wavyup](.05,1.2)node[anchor=west]{$d$};
} \label{eq:bubbleidentity}
& = \mfb_c(a,b) \: \tikz[trivalent]{\draw(-.05,-.2)to[wavyup](.05,1.2)node[rightlabel]{$c$};} \! \delta_{c,d}
\\
\tikz[trivalent]{
\draw(-.3,-.1)to[wavyup](-.2,.5)to[wavyup](-.3,1.1)node[leftlabel]{$a$}
(.3,-.1)to[wavyup](.2,.5)to[wavyup](.3,1.1)node[rightlabel]{$b$};
} \label{eq:fusionidentity}
\tikz[trivalent]{
\coordinate(vxa)at(0,.25)
edge[bend right]node[leftlabel,pos=1]{$a$}(-.3,-.15)edge[bend left]node[rightlabel,pos=1]{$b$}(.3,-.15);
\coordinate(vxb)at(0,.75)
edge[]node[basiclabel,auto]{$c$}(vxa)
edge[bend left]node[leftlabel,pos=1]{$a$}(-.3,1.15)edge[bend right]node[rightlabel,pos=1]{$b$}(.3,1.15);},
\end{align}
where $\mfb_c(a,b) = \frac{\Theta(a,b,c)}{\Delta(c)} = \frac{1}{\mff_c(a,b)}$.
\end{proposition}
We refer to $\mfb_c(a,b)$ and $\mff_c(a,b)$ as the \emph{bubble constant} and \emph{fusion constant}, respectively.

\subsection{$SL$-Equivariance and Trace Diagrams}
The rules for manipulating trace diagrams are precisely the same as those for manipulating spin networks, since all maps parts of a diagram are $\SL$-equivariant. In particular:
\begin{proposition}[Equivariance of Trace Diagrams, Propositions 3.8 and 3.11 in \cite{LP}]\label{s:diagram-equivarance}
For $g\in\SL$,
\begin{align*}
\tikz{\draw(0,.7)arc(180:360:.5);} &=
\tikz{\draw(0,1)--(0,.7)arc(180:360:.5)--(1,1);
\draw(0,.85)node[matrix]{$g$};\draw(1,.85)node[matrix]{$g$};}\\
\tikz[xscale=.7]{
\foreach\xa in{1,1.3,2.7,3}{\draw(\xa,0)to(\xa,1.3)node[small matrix,pos=.3]{$g$};}
\draw[draw=black,fill=gray!10](.8,.7)rectangle(3.2,1.1);\node[basiclabel]at(2,.9){$n$};
\node[basiclabel]at(2,.1){$\cdots$};
}&=\tikz[xscale=.7]{
\foreach\xa in{1,1.3,2.7,3}{\draw(\xa,0)to(\xa,1.3)node[small matrix,pos=.7]{$g$};}
\draw[draw=black,fill=gray!10](.8,.2)rectangle(3.2,.6);\node[basiclabel]at(2,.4){$n$};
\node[basiclabel]at(2,1.2){$\cdots$};
}\\
\tikz[trivalent]{\draw(0,.7)arc(180:360:.5)node[rightlabel]{$n$};} &=
\tikz[trivalent]{\draw(0,1)--(0,.7)arc(180:360:.5)--(1,1);\draw(.95,.4)node[rightlabel]{$n$};
\draw(0,.85)node[matrix]{$g$};\draw(1,.85)node[matrix]{$g$};}\\
\tikz[trivalent,shift={(0,1)},yscale=-1]{
\coordinate(vxa)at(0,.55)
edge[bend right]node[leftlabel,pos=1]{$a$}(-.6,-.4)edge[bend left]node[rightlabel,pos=1]{$b$}(.6,-.4)
edge[]node[rightlabel,pos=1]{$c$}(0,1.1);
\draw(-.5,0)node[matrix]{$g$};\draw(.5,0)node[matrix]{$g$};
} &=
\tikz[trivalent,shift={(0,1.3)},yscale=-1]{
\coordinate(vxa)at(0,.55)
edge[bend right]node[leftlabel,pos=1]{$a$}(-.6,-.1)edge[bend left]node[rightlabel,pos=1]{$b$}(.6,-.1)
edge[]node[rightlabel,pos=1]{$c$}(0,1.4);
\draw(0,1)node[matrix]{$g$};
}.
\end{align*}
\end{proposition}
Note the consequences
$$\tikz{\draw(0,.7)arc(180:360:.5)--(1,1);\draw(0,.85)node[matrix]{g};} = \tikz{\draw(0,1)--(0,.7)arc(180:360:.5)--(1,1); \draw(1,.85)node[small matrix,ellipse]{g^{-1}};} \qquad\text{and}\qquad \tikz[trivalent]{\draw(0,.7)arc(180:360:.5)--(1,1)node[rightlabel]{n};\draw(0,.85)node[matrix]{g};} = \tikz[trivalent]{\draw(0,1)--(0,.7)arc(180:360:.5)--(1,1);\draw(.95,.4)node[rightlabel]{n}; \draw(1,.85)node[small matrix,ellipse]{g^{-1}};}.$$

These equivariance properties permit every relation of the previous section to be adapted to include matrices along the edges. In later sections, we will often display diagrammatic manipulations without the required matrices, appealing to equivariance to show that they also apply with the matrices.


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