simplerecurrence.tex
\subsection{Simple Loop Recurrences}\label{simplerecurrence-section}\label{s:simple}

This section describes how to apply Proposition \ref{l:vertexglue2} to generate a product relation for each simple loop in a trace diagram. These will translate directly to recurrences of central functions.

%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{The Main Results}
First, \eqref{eq:gluevertexleft} can be ``stacked'' to obtain a more general formula for gluing across two or more vertices.

\begin{lemma}\label{l:vertexglue2}
    If the edges labeled by $\{a_0,\ldots,a_n\}$ are disjoint, then
        $$
        \tikz[trivalent,shift={(0,-2)}]{
            \foreach\xa/\xb/\xc in{0/0/a_0,-.1/.8/a_1,-.4/3/a_{n-1},-.5/3.8/a_n}{
                \draw(\xa,\xb)to[out=110,in=-90]([shift={(\xa,\xb)}]-.1,.8)node[rightlabel,pos=.8]{$\xc$};
                \draw[blue,shift={(-.35,0)}](\xa,\xb)to[wavyup]([shift={(\xa,\xb)}]-.1,.8);
            }
            \foreach\xa/\xb/\xc in{-.1/.8/b_1,-.5/3.8/b_n}{
                \draw(\xa,\xb)to[bend right]([shift={(\xa,\xb)}].6,.7)node[rightlabel]{$\xc$};
            }
            \draw[dotted](-.2,1.6)to[wavyup](-.4,3);
            \draw[blue,dotted,shift={(-.35,0)}](-.2,1.6)to[wavyup](-.4,3);
            \node[blue,leftlabel]at(-1,4.6){$1$};
        }
        = \sum_{\substack{a_0'\in\iadm{1,a_0}\\ \{a_i'\in\iadm{1,a_i}\cap\iadm{b_i,a_{i-1}'}\}_{i=1}^n}}
            \prod_{i=1}^n\mfs_{a_{i-1}'}(a_{i-1},1)\nFus{b_i}{a_{i-1}}{a_{i-1}'}{a_i}{a_i'}
        \tikz[trivalent,shift={(0,-2)}]{
            \foreach\xa/\xb/\xc in{-.1/.8/a_1',-.4/3/a_{n-1}'}{
                \draw(\xa,\xb)to[out=110,in=-90]([shift={(\xa,\xb)}]-.1,.8)node[rightlabel,pos=.8]{$\xc$};}
            \foreach\xa/\xb/\xc in{-.1/.8/b_1,-.5/3.8/b_n}{
                \draw(\xa,\xb)to[bend right]([shift={(\xa,\xb)}].6,.7)node[rightlabel]{$\xc$};}
            \draw[dotted](-.2,1.6)to[wavyup](-.4,3);
            \coordinate(vxa)at(0,.3)
                edge[wavyup]node[rightlabel]{$a_0'$}(-.1,.8)
                edge[blue,bend right]node[leftlabel,pos=1]{$1$}(-.3,-.1)
                edge[bend left]node[rightlabel,pos=1]{$a_0$}(.2,-.1);
            \coordinate(vxb)at(-.6,4.3)
                edge[wavydown]node[leftlabel]{$a_n'$}(-.5,3.8)
                edge[blue,bend left]node[leftlabel,pos=1]{$1$}(-.9,4.7)
                edge[bend right]node[rightlabel,pos=1]{$a_n$}(-.4,4.7);
        }.
        $$
\begin{proof}
    For clarity, we present the concrete two-vertex case here, from which the general pattern can be seen. First, apply \eqref{eq:gluevertexleft} on the upper and lower halves of the diagram separately:
    \begin{multline*}
    \tikz[trivalent,shift={(0,-.3)}]{
        \draw[very thick,lightgray](-.15,.5)to(1.45,.5);
        \foreach\xx/\xy/\xa/\xb/\xcc/\xd in{.25/-.6/a_0/b_1//,-.25/.6/a_1/b_2/a_2/1}{\begin{scope}[shift={(\xx,\xy)}]
        \coordinate(vxa)at(.5,.5)
            edge[]node[rightlabel,pos=.45]{$\xa$}(.5,-.1)
            edge[bend left]node[rightlabel,pos=1]{$\xcc$}(0,1.1)
            edge[bend right]node[rightlabel,pos=.7]{$\xb$}(1,1.1);
        \draw[blue](.3,-.1)to[wavyup](-.2,1.1)node[leftlabel]{$\xd$};
        \end{scope}}
    }
    = \sum_{\substack{a_0'\in\iadm{1,a_0}\\
        a_1'\in\iadm{1,a_1}\cap\iadm{b_1,a_0'}\\
        a_2'\in\iadm{1,a_2}\cap\iadm{b_2,a_1'}}}
    \mfs_{a_0'}(a_0,1)\Fus{b_1}{a_0}{a_0'}{a_1}{a_1'}
    \mfs_{a_1'}(a_1,1)\Fus{b_2}{a_1}{a_1'}{a_2}{a_2'}
    \tikz[trivalent]{
        \draw[very thick,lightgray](-.65,.3)to(.95,.3);
        \foreach\xx/\xy/\xa/\xb/\xcc/\xd in{.15/-.85/a_0/b_1/a_1/1,-.15/.85/a_1/b_2/a_2/1}
            {\begin{scope}[shift={(\xx,\xy)}]
        \coordinate(vxa)at(0,.35)
            edge[bend right]node[rightlabel,pos=.7]{$\xb$}(.6,1.15);
        \coordinate(vxc)at(0,-.15)
            edge[]node[leftlabel]{$\xa'$}(vxa)
            edge[blue,bend right]node[leftlabel,pos=1]{$\xd$}(-.3,-.55)
            edge[bend left]node[rightlabel,pos=1]{$\xa$}(.3,-.55);
        \coordinate(vxb)at(-.3,.75)
            edge[bend right]node[pos=.7,pin={[basiclabel]left:$\xcc'$}]{}(vxa)
            edge[bend right]node[rightlabel,pos=1]{$\xcc$}(0,1.15)
            edge[blue,bend left]node[leftlabel,pos=1]{$\xd$}(-.6,1.15);
        \end{scope}}}
    \\
    = \sum_{\substack{a_0'\in\iadm{1,a_0}\\ \{a_i'\in\iadm{1,a_i}\cap\iadm{b_i,a_{i-1}'}\}_{i=1}^2}}
    \mfs_{a_0'}(a_0,1)\Fus{b_1}{a_0}{a_0'}{a_1}{a_1'}
    \mfs_{a_1'}(a_1,1)\Fus{b_2}{a_1}{a_1'}{a_2}{a_2'} \mfb_{a_1'}(1,a_1)
    \tikz[trivalent,scale=.9,shift={(0,-.2)}]{
        \coordinate(vxa)at(0,.35)
            edge[bend right]node[rightlabel,pos=.7]{$b_1$}(.6,1.15)
            edge[bend left](-.3,1.15);
        \coordinate(vxc)at(0,-.15)
            edge[]node[leftlabel]{$a_0'$}(vxa)
            edge[blue,bend right]node[leftlabel,pos=1]{$1$}(-.3,-.55)
            edge[bend left]node[rightlabel,pos=1]{$a_0$}(.3,-.55);
        \begin{scope}[shift={(-.3,1.7)}]
        \coordinate(vxa)at(0,.35)
            edge[bend right]node[rightlabel,pos=.7]{$b_2$}(.6,1.15)
            edge[]node[leftlabel]{$a_1'$}(0,-.7);
        \coordinate(vxb)at(-.3,.75)
            edge[bend right]node[pos=.7,pin={[basiclabel]left:$a_2'$}]{}(vxa)
            edge[bend right]node[rightlabel,pos=1]{$a_2$}(0,1.15)
            edge[blue,bend left]node[leftlabel,pos=1]{$1$}(-.6,1.15);
        \end{scope}
    }
    \end{multline*}
    Note that the bubble identity \eqref{eq:bubbleidentity} allows us to use the same index $a_1'$ above and below the bubble in the second diagram and contributes the $\mfb_{a_1'}(1,a_1)$ term in the second step. Using \eqref{eq:fusnormalization} and the fact that $\mff_c(a,b)\mfb_c(a,b)=1$, the coefficient can be expressed as
    $$\sqrt{\mff_{a_0'}(a_0,1)\mff_{a_2'}(a_2,1)}
        \prod_{i=1}^2\mfs_{a_i'}(a_i,1)\nFus{b_i}{a_{i-1}}{a_{i-1}'}{a_i}{a_i'}.
    $$
    The case with more than two vertices is similar.
\end{proof}
\end{lemma}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We now introduce notation that describes relabelling of spin networks. In what follows, we let $\mathsf{s}$ be a closed trivalent graph with edges $E=(e_1,e_2,\ldots,e_m)$. Given a labelling $l:E\to\mathbb{N}$ of the edges by natural numbers such that the labels at all vertices form admissible triples, let $\mathsf{s}_l$ denote the resulting spin network.

\begin{definition}
    An \emph{admissible relabelling} of $l$ is a labelling $l_a$ of a subset of edges $a:(e_{j_1},e_{j_2},\ldots,e_{j_n})\to\mathbb{N}$ such that the resulting network $\mathsf{s}_{l_a}$ is still admissible.
\end{definition}

The next theorem says that multiplication of a spin network by a simple loop may be expanded in terms of diagrams with admissible relabellings.

\begin{theorem}[Simple Loop Multiplication Formula]\label{t:simplerecurrence}
    Let $\mathsf{s}$ be a trivalent graph with admissible labelling $l:(e_1,e_2,\ldots,e_m)\to\mathbb{N}$. Let $\gamma$ denote both a simple cycle $(e_{j_1},e_{j_2},\ldots,e_{j_n})$ of $\mathsf{s}$ and the corresponding spin network. Define $a_i=l(e_{j_i})$, and let $b_i\in\iadm{a_{i-1},a_i}$ be the third label on the vertex joining edges $e_{j_{i-1}}$ and $e_{j_i}$. Let $a':e_{j_i}\mapsto a_i'$ be an admissible relabelling of $\mathsf{s}_l$.
    With the understanding that $e_{j_0}=e_{j_n}$ and $a_0=a_n$,
    $$\gamma\cdot\mathsf{s}_l
     =\sum_{\{a_i'\in\iadm{1,a_i}\cap\iadm{b_i,a_{i-1}'}\}_{i=1}^n}
        \left(\prod_{i=1}^n\mfs_{a_i'}(1,a_i)^{N_i}\nFus{b_i}{a_{i-1}}{a_{i-1}'}{a_i}{a_i'}\right)
        \mathsf{s}_{l_{a'}},$$
    where $N_i$ is a number related to the topology of $\gamma$ within the diagram; in particular, it is the sum of (i) the number of times $\gamma$ crosses the edge $e_{j_i}$, (ii) the number of local extrema along the edge $e_{j_i}$ that do not occur at a vertex, and (iii) the number of times $e_{j_i}$ adjoins a vertex as shown in Figure \ref{f:ni-definition}.
    \begin{figure}[htb]
    \begin{tabular}{ccccc}
    \tikz[trivalent]{
        \draw(-.1,-.2)to[wavyup](-.2,.2)to[wavyup](.2,.8)to[wavyup](.1,1.2)node[rightlabel]{$a_i$};
        \draw[blue](.1,-.2)to[wavyup](.2,.2)to[wavyup](-.2,.8)to[wavyup](-.1,1.2)node[leftlabel]{$1$};
    }
    & &
    \tikz[trivalent]{
        \draw(-.5,.8)node[leftlabel]{$a_i$}to[out=-90,in=-90,looseness=3](.5,.8);
        \draw[blue](-.3,.8)node[rightlabel]{$1$}to[out=-90,in=-90,looseness=3](.3,.8);
    }\:\:\:
    & &
    \tikz[trivalent]{
        \coordinate(vxa)at(.5,.5)
            edge[]node[rightlabel,pos=1]{$a_{i\pm1}$}(.5,-.1)
            edge[bend left]node[rightlabel,pos=1]{$a_i$}(0,1.1)
            edge[bend right]node[rightlabel,pos=1]{$b_{i\pm1}$}(1,1.1);
        \draw[blue](.3,-.1)to[wavyup](-.2,1.1)node[leftlabel]{$\gamma$};}
    \tikz[trivalent]{
        \coordinate(vxa)at(.5,.5)
            edge[]node[leftlabel,pos=1]{$a_{i\pm1}$}(.5,-.1)
            edge[bend left]node[leftlabel,pos=1]{$b_{i\pm1}$}(0,1.1)
            edge[bend right]node[leftlabel,pos=1]{$a_i$}(1,1.1);
        \draw[blue](.7,-.1)to[wavyup](1.2,1.1)node[rightlabel]{$\gamma$};}
    \tikz[trivalent]{
        \coordinate(vxa)at(.5,.5)
            edge[]node[rightlabel,pos=1]{$a_{i\pm1}$}(.5,1.1)
            edge[bend left]node[rightlabel,pos=1]{$b_{i\pm1}$}(1,-.1)
            edge[bend right]node[rightlabel,pos=1]{$a_i$}(0,-.1);
        \draw[blue](-.2,-.1)to[wavyup](.3,1.1)node[leftlabel]{$\gamma$};}
    \tikz[trivalent]{
        \coordinate(vxa)at(.5,.5)
            edge[]node[leftlabel,pos=1]{$a_{i\pm1}$}(.5,1.1)
            edge[bend left]node[leftlabel,pos=1]{$a_i$}(1,-.1)
            edge[bend right]node[leftlabel,pos=1]{$b_{i\pm1}$}(0,-.1);
        \draw[blue](1.2,-.1)to[wavyup](.7,1.1)node[rightlabel]{$\gamma$};
    }
    \\
    \textrm{(a)} & & \textrm{(b)} & & \textrm{(c)}
    \end{tabular}
    \caption{Local contributors to sign of simple loop multiplication formula.}\label{f:ni-definition}
    \end{figure}
\begin{proof}
    First, suppose the loop has the form of Lemma \ref{l:vertexglue2}, with the $a_0$ and $a_n$ edges coinciding, as follows:
    $$
        \tikz[trivalent,shift={(0,.2)}]{
            \draw[blue](-.2,.8)arc(0:180:.35)--(-.9,-.8)arc(180:360:.35)--cycle;
            \draw(0,.8)arc(0:180:.55)--(-1.1,-.8)arc(180:360:.55)--cycle;
            \draw(0,-.4)to[bend right](.5,0);\draw(0,.4)to[bend right](.5,.8);
            \node at(.25,-.1){.};\node at(.25,.1){.};\node at(.25,.3){.};
        }
        \longrightarrow
        \tikz[trivalent,shift={(0,.2)}]{
            \draw[blue](0,.6)to[bend left](-.2,.8)arc(0:180:.35)--(-.9,-.8)arc(180:360:.35)to[bend left](0,-.6);
            \draw(0,.8)arc(0:180:.55)--(-1.1,-.8)arc(180:360:.55)--cycle;
            \draw(0,-.4)to[bend right](.5,0);\draw(0,.4)to[bend right](.5,.8);
            \node at(.25,-.1){.};\node at(.25,.1){.};\node at(.25,.3){.};
        }
        \longrightarrow
        \tikz[trivalent,shift={(0,.2)}]{
            \draw(0,.8)arc(0:180:.55)--(-1.1,-.8)arc(180:360:.55)--cycle;
            \draw(0,-.4)to[bend right](.5,0);\draw(0,.4)to[bend right](.5,.8);
            \node at(.25,-.1){.};\node at(.25,.1){.};\node at(.25,.3){.};
        }
    $$
    Popping the final bubble introduced along these edges results in an additional factor of $\mfb_{a_0'}(a_0,1)$, which cancels with the $\sqrt{\mff_{a_0'}(a_0,1)\mff_{a_n'}(a_n,1)}=\mff_{a_0'}(a_0,1)$ term. So in this case the  coefficients of the summation are
        $$\prod_{i=1}^n \mfs_{a_i'}(1,a_i) \nFus{b_i}{a_{i-1}}{a_{i-1}'}{a_i}{a_i'}.$$

    In general, the relative positions of the labels $\{a_{i-1},a_i,b_i\}$ in the vicinity of a vertex matters. If the product appears locally at a local extrema, as in \eqref{eq:glueonesum2} or its reflection, then the term $\mfs_{a_i'}(1,a_i)$ is unnecessary. Otherwise, in the cases depicted in Figure \ref{f:ni-definition}(c), the term remains.

    If a crossing or local extrema occurs along an edge, then the ``bubble popping'' step in Lemma \ref{l:vertexglue2} becomes one of the following:
    \begin{align*}
    \tikz[trivalent]{
        \draw(-.15,-.5)to[wavyup](-.1,-.2)to[wavyup](-.2,.2)to[wavyup](.2,.8)to[wavyup](.1,1.2)to[wavyup](.15,1.5)node[rightlabel]{$a_i$};
        \draw[blue](.15,-.5)to[wavyup](.1,-.2)to[wavyup](.2,.2)to[wavyup](-.2,.8)to[wavyup](-.1,1.2)to[wavyup](-.15,1.5)node[leftlabel]{$1$};}
    :\qquad
    \tikz[trivalent]{
        \coordinate(vxa)at(0,-.4)edge[bend right]node[leftlabel,pos=1]{$a_i$}(-.3,-.6)edge[blue,bend left]node[rightlabel,pos=1]{$1$}(.3,-.6);
        \coordinate(vxb)at(0,.15)edge[]node[rightlabel]{$a_i'$}(vxa)edge[bend left](-.3,.25)edge[blue,bend right](.3,.25);
        \coordinate(vxd)at(0,1.4)edge[blue,bend left]node[leftlabel,pos=1]{$1$}(-.3,1.6)edge[bend right]node[rightlabel,pos=1]{$a_i$}(.3,1.6);
        \coordinate(vxc)at(0,.85)edge[]node[rightlabel]{$b_i'$}(vxd)edge[bend left](.3,.75)edge[blue,bend right](-.3,.75);
        \draw(-.3,.25)node[leftlabel]{$a_i$}to[wavyup](.3,.75);
        \draw[blue](.3,.25)node[rightlabel]{$1$}to[wavyup](-.3,.75);
    }
    &=\mfs_{a_i'}(1,a_i)\mfb_{a_i'}(1,a_i)
    \tikz[trivalent]{
        \coordinate(vxa)at(0,.25)
            edge[bend right]node[leftlabel,pos=1]{$a_i$}(-.3,-.2)
            edge[blue,bend left]node[rightlabel,pos=1]{$1$}(.3,-.2);
        \coordinate(vxb)at(0,.75)
            edge[]node[auto,basiclabel]{$a_i'$}(vxa)
            edge[blue,bend left]node[leftlabel,pos=1]{$1$}(-.3,1.2)
            edge[bend right]node[rightlabel,pos=1]{$a_i$}(.3,1.2);};\\
    \tikz[trivalent]{
        \draw(-.5,.8)node[leftlabel]{$a_i$}to[out=-90,in=-90,looseness=3](.5,.8);
        \draw[blue](-.3,.8)node[rightlabel]{$1$}to[out=-90,in=-90,looseness=3](.3,.8);
    }
    :\qquad
    \tikz[trivalent]{
        \coordinate(vxa)at(-.5,.7)edge[bend left]node[leftlabel,pos=1]{$a_i$}(-.8,1.1)edge[blue,bend right]node[leftlabel,pos=1]{$1$}(-.2,1.1);
        \coordinate(vxc)at(-.5,.3)edge[]node[leftlabel]{$a_i'$}(vxa)edge[bend right]node[leftlabel,pos=1]{$a_i$}(-.8,0)edge[bend left,blue](-.2,0);
        \coordinate(vxb)at(.5,.7)edge[bend right]node[rightlabel,pos=1]{$a_i$}(.8,1.1)edge[blue,bend left]node[rightlabel,pos=1]{$1$}(.2,1.1);
        \coordinate(vxd)at(.5,.3)edge[]node[rightlabel]{$a_i'$}(vxb)edge[bend left](.8,0)edge[bend right,blue](.2,0);
        \draw(-.8,0)to[out=-90,in=-90](.8,0);
        \draw[blue](-.2,0)to[out=-90,in=-90]node[basiclabel,auto]{$1$}(.2,0);
    }
    &=\mfs_{a_i'}(1,a_i)\mfb_{a_i'}(1,a_i)
    \tikz[trivalent]{
        \coordinate(vxa)at(-.5,.7)edge[bend left]node[leftlabel,pos=1]{$a_i$}(-.8,1.1)edge[blue,bend right]node[leftlabel,pos=1]{$1$}(-.2,1.1);
        \coordinate(vxb)at(.5,.7)edge[bend right]node[rightlabel,pos=1]{$a_i$}(.8,1.1)edge[blue,bend left]node[rightlabel,pos=1]{$1$}(.2,1.1);
        \draw(vxa)to[out=-90,in=-90,looseness=3]node[basiclabel,auto]{$a_i'$}(vxb);
    }.
    \end{align*}
    The signs are calculated using Lemma \ref{p:spinnetsigns} and the stronger Proposition \ref{p:spinnetsignstrong}. In the second case, the sign is calculated by comparing the $\mfe_{a_i'}(1,a_i)$ kinks in the diagram
    \tikz[trivalent,scale=.75,shift={(0,.4)}]{
        \coordinate(vxc)at(-.5,.3)
            edge[]node[leftlabel]{$a_i'$}(-.5,.8)
            edge[bend right](-.7,.1)
            edge[bend left,blue](-.2,.1);
        \draw(-.7,.1)arc(180:360:.7)--(.7,.8)node[rightlabel,auto]{$a_i$};
        \draw[blue](-.2,.1)arc(180:360:.25)--(.3,.8)node[leftlabel,auto]{$1$};
    }
    with the diagram
    \tikz[trivalent,scale=.75,shift={(0,.3)}]{
        \draw(-.5,.8)node[leftlabel]{$a_i'$}--(-.5,.2)arc(180:360:.5);
        \draw(.5,.2)to[bend left,blue](.3,.8)node[leftlabel,auto]{$1$};
        \draw(.5,.2)to[bend right](.7,.8)node[rightlabel,auto]{$a_i$};
    },
    which has no kinks.
    In each case the additional sign $\mfs_{a_i'}(1,a_i)$ adds one to the exponent $N_i$.
\end{proof}
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%
Rearranging the terms in the above theorem provides a recurrence formula, in which each diagram can be written in terms of diagrams of lower rank, where rank is defined as follows.
\begin{definition}
The \emph{rank} of a spin network $\mathsf{s}_l$ is the sum $\sum_{i=1}^m l(e_i)$.
\end{definition}
\begin{corollary}\label{c:simplerecurrence}
    Let $\mathsf{s}$, $l$, $\gamma$, $a_i$, $b_i$, $N_i$, and $a'$ be defined as in Theorem \ref{t:simplerecurrence}. If $\mathsf{s}_{l_{\na}}$ is an admissible partial relabelling, where $\na(e_{j_i})=l(e_{j_i})-1$, then $\mathsf{s}_l$ can be expressed in terms of $\gamma$ and spin networks of lower rank:
    \begin{equation}\label{eq:simplerecurrence}
        \mathsf{s}_l = \gamma\cdot\mathsf{s}_{l_{\na}}
        - \sum_{\substack{\{a_i'\in\iadm{1,a_i-1}\cap\iadm{b_i,a_{i-1}'}\}_{i=1}^n\\ \text{some }a_i'\ne a_i}}
        \left(\prod_{i=1}^n\mfs_{a_i'}(1,a_i-1)^{N_i}\nFus{b_i}{a_{i-1}-1}{a_{i-1}'}{a_i-1}{a_i'}\right)
        \mathsf{s}_{l_{a'}}.
    \end{equation}
\begin{proof}
    The term with $a'(e_{j_i})=\pa_i$ in Theorem \ref{t:simplerecurrence} is the unique one with highest rank. So one may re-index by replacing the $a_i$ with $a_i-1$ as shown. Since $\mfs_{a_i}(1,a_i-1)=+1$ and $\nFus{b_i}{a_{i-1}-1}{a_{i-1}}{a_i-1}{a_i}=1$, there is no coefficient on the $\mathsf{s}_l$ term.
\end{proof}
\end{corollary}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Some Examples}

We illustrate the application of these theorems in a few basic examples. Note that the sign is only a factor when $a_i'=a_i+1$ since
    $$\mfs_{a_i'}(1,a_i)=
    \begin{cases}
        +1  &   a_i'=a_i+1  \\
        -1  &   a_i'=a_i-1.
    \end{cases}$$
We will give the recurrences in their most general form; for some choices of labels, the non-admissible terms should be excluded.

\begin{example}\label{ex:loopone}
    A single edge loop has a two-term recurrence:
    \begin{equation}\label{eq:loopone}
    \begin{matrix}
    \tikz[trivalent]{
        \draw[blue](0,.5)circle(.4);\node[blue,basiclabel]at(.1,.5){$\gamma$};
        \draw(0,.5)circle(.6);\node[basiclabel]at(.7,0){$a$};
        \draw(0,1.1)to[bend left](.1,1.5)node[rightlabel]{$b$};}
     &=& \nFus ba{a+1}a{a+1}
     &\tikz[trivalent]{\draw(0,.5)circle(.5);\node[basiclabel]at(.7,-.1){$a+1$};\draw(0,1)to[bend left](.1,1.4)node[rightlabel]{$b$};}
     &-& \nFus ba{a-1}a{a-1}
     &\tikz[trivalent]{\draw(0,.5)circle(.5);\node[basiclabel]at(.7,-.1){$a-1$};\draw(0,1)to[bend left](.1,1.4)node[rightlabel]{$b$};}
     \\
     &=&
     &\tikz[trivalent]{\draw(0,.5)circle(.5);\node[basiclabel]at(.7,-.1){$a+1$};\draw(0,1)to[bend left](.1,1.4)node[rightlabel]{$b$};}
     &+&\tfrac{(a-\frac b2)(a+\frac b2+1)}{a(a+1)}
     &\tikz[trivalent]{\draw(0,.5)circle(.5);\node[basiclabel]at(.7,-.1){$a-1$};\draw(0,1)to[bend left](.1,1.4)node[rightlabel]{$b$};}.
    \end{matrix}
    \end{equation}
    The loop $\gamma$ consists of a single edge, with $N_1=1$ since there are no crossings and $\gamma$ has one extremum that does not occur at a vertex.
\end{example}
Setting $b=0$, one obtains a formula equivalent to \ref{eq:rank1recurrence}, which can be used to compute the rank one central functions.

\begin{example}\label{ex:looptwo}
    A two-edge loop has the following four-term recurrence:
    \begin{equation}\label{eq:looptwo}
    \begin{matrix}
    \tikz[trivalent]{
        \draw[blue](0,.5)circle(.4);\node[blue,basiclabel]at(.1,.5){$\gamma$};
        \draw(0,.5)circle(.6);\node[basiclabel]at(-.85,.8){$a_1$};\node[basiclabel]at(.85,.2){$a_2$};
        \draw(0,1.1)to[bend left](.1,1.5)node[rightlabel]{$b_2$};
        \draw(0,-.1)to[bend left](-.1,-.5)node[leftlabel]{$b_1$};}
     &=&
     \displaystyle{\sum_{\substack{a_1'=a_1\pm1\\ a_2'=a_2\pm1}}
        \left(\prod_{i=1}^2 \nFus{b_i}{a_1}{a_1'}{a_2}{a_2'}\right)}
     \tikz[trivalent]{
        \draw(0,.5)circle(.5);\node[basiclabel]at(-.8,.7){$a_1'$};\node[basiclabel]at(.8,.3){$a_2'$};
        \draw(0,1)to[bend left](.1,1.4)node[rightlabel]{$b_2$};
        \draw(0,0)to[bend left](-.1,-.4)node[leftlabel]{$b_1$};}
    \end{matrix}
    \end{equation}
    Since $N_1=0+0+0=N_2$, no additional signs are necessary.
\end{example}

\subsubsection{Application to Rank Two Central Functions}
The rank two central function is
    $$
    \ch cab (\xb_1,\xb_2) =
    \tikz[trivalent,every node/.style={basiclabel}]{
        \draw(0,.5)circle(.4)(0,.1)arc(-145:145:.7);
        \node[small matrix]at(-.4,.5){$\mathbf{X}_1$};
        \node[small matrix]at(.4,.5){$\mathbf{X}_2$};
        \node at(-.4,1){$a$};\node at(.5,.95){$b$};\node at(1.2,1.1){$c$};
    }
    $$
There are three simple loops: $(a,b)$ corresponding to $\Tr{\xb_1\xb_2^{-1}}$, $(a,c)$ corresponding to $\Tr{\xb_1}$, and $(b,c)$ corresponding to $\Tr{\xb_2}$. Each loop provides a different recurrence, a fact which was used in \cite{LP} to obtain a new proof of a classical theorem of Fricke, Klein, and Vogt.

For example, closing off each term in \eqref{ex:looptwo} gives the recurrence
    \begin{multline}\label{eq:fourtermcf2}
    \tikz[scale=1.5,yscale=.7]{
        \draw[trivalent]
            (0,0)to[bend left=80](0,1)node[leftlabel,pos=.75]{$a$}
            (0,0)to[bend right=80](0,1)node[rightlabel,pos=.75]{$b$};
        \draw[trivalent]
            (0,0)to[bend right](.5,-.3)
            (.5,-.3)to[bend right=80](.5,1.3)node[rightlabel,pos=.7]{$c$}
            (.5,1.3)to[bend right=50](0,1);
        \draw[blue,trivalent]
            (0,.15)to[bend left=80](0,.85)to[bend left=80](0,.15)node[leftlabel,pos=.5]{$\gamma$};
    }
    =
    \tikz[scale=1.5,yscale=.7]{
        \draw[trivalent]
            (0,0)to[bend left=80](0,1)node[leftlabel,pos=.75,scale=.75]{$a+1$}
            (0,0)to[bend right=80](0,1)node[rightlabel,pos=.75,scale=.75]{$b+1$};
        \draw[trivalent]
            (0,0)to[bend right](.5,-.3)
            (.5,-.3)to[bend right=80](.5,1.3)node[rightlabel,pos=.7,scale=.75]{$c$}
            (.5,1.3)to[bend right=50](0,1);
    }
    +\tfrac{\mfe_a(b,c)^2}{b(b+1)}
    \tikz[scale=1.5,yscale=.7]{
        \draw[trivalent]
            (0,0)to[bend left=80](0,1)node[leftlabel,pos=.75,scale=.75]{$a+1$}
            (0,0)to[bend right=80](0,1)node[rightlabel,pos=.75,scale=.75]{$b-1$};
        \draw[trivalent]
            (0,0)to[bend right](.5,-.3)
            (.5,-.3)to[bend right=80](.5,1.3)node[rightlabel,pos=.7,scale=.75]{$c$}
            (.5,1.3)to[bend right=50](0,1);
    }\\
    +\tfrac{\mfe_b(a,c)^2}{a(a+1)}
    \tikz[scale=1.5,yscale=.7]{
        \draw[trivalent]
            (0,0)to[bend left=80](0,1)node[leftlabel,pos=.75,scale=.75]{$a-1$}
            (0,0)to[bend right=80](0,1)node[rightlabel,pos=.75,scale=.75]{$b+1$};
        \draw[trivalent]
            (0,0)to[bend right](.5,-.3)
            (.5,-.3)to[bend right=80](.5,1.3)node[rightlabel,pos=.7,scale=.75]{$c$}
            (.5,1.3)to[bend right=50](0,1);
    }
    +\tfrac{\mfe_c(a,b)^2(\mfe(a,b,c)+1)^2}{a(a+1)b(b+1)}
    \tikz[scale=1.5,yscale=.7]{
        \draw[trivalent]
            (0,0)to[bend left=80](0,1)node[leftlabel,pos=.75,scale=.75]{$a-1$}
            (0,0)to[bend right=80](0,1)node[rightlabel,pos=.75,scale=.75]{$b-1$};
        \draw[trivalent]
            (0,0)to[bend right](.5,-.3)
            (.5,-.3)to[bend right=80](.5,1.3)node[rightlabel,pos=.7,scale=.75]{$c$}
            (.5,1.3)to[bend right=50](0,1);
    }.
    \end{multline}
Note the similarity to the formula in Theorem \ref{t:ranktworecurrencex}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Application to Rank Three Central Functions}

The left-associative rank three central functions are
    $$
    \chi_{a,b,c,d,e,f}=
    \tikz[scale=1.4]{
        \draw[trivalent]
            (0,0)to[bend left=80]node[small matrix]{$\xb_1$}(0,1)node[leftlabel,pos=.8]{$a$}
            (0,0)to[bend right=80]node[small matrix]{$\xb_2$}(0,1)node[rightlabel,pos=.8]{$b$}
            (0,0)to[bend right=20](.5,-.2)node[bottomlabel,pos=.5]{$e$}
            to[bend right=80]node[small matrix]{$\xb_3$}(.5,1.2)node[rightlabel,pos=.75]{$c$}
            to[bend right=20](0,1)node[toplabel,pos=.5]{$f$}
            (.5,-.2)to[bend right=20](1,-.4)
            to[bend right=80](1,1.4)node[rightlabel,pos=.75]{$d$}
            to[bend right=20](.5,1.2);
    }
    $$
There are six simple loops in the diagram:
    \begin{align*}
        (a,b) \leftrightarrow \Tr{\xb_1\xb_2^{-1}}, &\qquad (c,d) \leftrightarrow \Tr{\xb_3}, \\
        (a,e,d,f) \leftrightarrow \Tr{\xb_1}, &\qquad (b,e,d,f) \leftrightarrow \Tr{\xb_2}, \\
        (a,e,c,f) \leftrightarrow \Tr{\xb_1\xb_3^{-1}}, &\qquad (b,e,c,f) \leftrightarrow \Tr{\xb_2\xb_3^{-1}}.
    \end{align*}
However, as noted in Section \ref{s:introduction}, up to seven variables may be required in the expansion of rank three central functions, so the simple terms do not suffice to compute all central functions. This case will be treated in detail in section \ref{ss:rank3combinatorial}.

\ifnum0=1
    $$\chi_{a_i,c_i}=x_i\cdot\chi_{a_i-1,c_i-1}
            -(-1)^{\beta_i^1+\beta_i^2}\left(\frac{\alpha_i^1\alpha_i^2}{c_i(c_i+1)}\chi_{a_i-2,c_i}
                +\frac{\gamma_i^1\gamma_i^2}{a_i(a_i+1)}\chi_{a_i,c_i-2}\right)
            -\frac{\beta_i^1\beta_i^2(\delta_i^1+1)(\delta_i^2+1)}{a_i(a_i+1)c_i(c_i+1)}\chi_{a_i-2,c_i-2}.$$
\fi

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Application to General Central Functions}
In section \ref{s:centralfunctions}, the left associative central functions were shown to be given by
    $$
    \chi_{\vi,\vec m,\vec p} \equiv
    \tikz[scale=1.2]{
        \draw[trivalent]
            (0,0)to[bend left=80]node[small matrix]{$X_1$}(0,1)node[leftlabel,pos=.8]{$i_1$}
            (0,0)to[bend right=80]node[small matrix]{$X_2$}(0,1)node[rightlabel,pos=.8]{$i_2$}
            (0,0)to[bend right=20](.5,-.2)node[bottomlabel,pos=.5]{$m_1$}
            to[bend right=80]node[small matrix]{$X_3$}(.5,1.2)node[rightlabel,pos=.8]{$i_3$}
            to[bend right=20](0,1)node[toplabel,pos=.5]{$p_1$}
            (.5,-.2)to[bend right=20](1,-.4)node[bottomlabel,pos=.5]{$m_2$}
            (1,1.4)to[bend right=20](.5,1.2)node[toplabel,pos=.5]{$p_2$};
        \draw[draw=none](1.25,0)--(2.25,-.2)node[pos=.2]{.}node[pos=.5]{.}node[pos=.8]{.};
        \draw[draw=none](1.25,1)--(2.25,1.2)node[pos=.2]{.}node[pos=.5]{.}node[pos=.8]{.};
        \draw[trivalent,shift={(.5,0)}]
            (1.5,-.6)to[bend right=20](2,-.8)node[bottomlabel,pos=.4]{$m_{r-2}$}
            to[bend right=80]node[small matrix]{$X_r$}(2,1.8)node[leftlabel,pos=.8]{$i_r$}
            to[bend right=20](1.5,1.6)node[toplabel,pos=.6]{$p_{r-2}$}
            (2,-.8)to[bend right=20](2.5,-1)
            to[bend right=80](2.5,2)node[rightlabel,pos=.4]{$m_{r-1}$}
            to[bend right=20](2,1.8);
    }.
    $$

\begin{theorem}
    There are $\frac{r(r+1)}{2}$ possible simple loops in the general rank $r$ left-associative central function. Multiplication by any of these possible loops gives rise to a recurrence as described in Theorem \ref{t:simplerecurrence}.
\begin{proof}
    Simple loops must pass through precisely two of the edges labeled by $i_l$, implying that there are $\binom{r+1}{2}=\frac{r(r+1)}{2}$ possible simple loops.
\end{proof}
\end{theorem}
The simple recurrences given in this way correspond to the subset of $\C[\X_r]$ generators containing $\mathcal{G}_1=\{\Tr{\xb_1},...,\Tr{\xb_r}\}$ and $\mathcal{G}_2=\{\Tr{\xb_i\xb_j}\ |\ 1\leq i<j \leq r\}$.

%Note that the recurrences may have a large number of terms. In particular, if $1\leq i<j\leq r-1$, the loop through $d_i,d_j$ will have $2+2(j-i)=2(j-i+1)$ segments. If $i=0$ or $j=r$, this formula should be reduced by two, so the number of terms in the recurrence is:
% $$
% \begin{cases}
%    4^{j-i+1} & 1\leq i<j\leq r-1\\
%    4^{j-i} & 0=i<j\leq r-1 \quad\text{or}\quad 1\leq i<j=r\\
%    4^{j-i-1} & 0=i<j=r.
% \end{cases}
% $$
%While this number is large, the computation of the coefficients is straightforward and easily implemented using a computer algebra system.
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