\appendix

\section{A brief note on $T_n$, a related planar algebra}
\label{appendix}
In this section we briefly describe modifications of the skein relations
for $D_{2n}$ which give rise to the planar algebras $T_n$. The planar algebras $T_n$ have appeared previously in \cite{MR1936496, MR2046203, MR1333750}.
They are \emph{unshaded} subfactor planar algebras in the sense
we've described in \ref{subsec:pa}, but they are not \emph{shaded} subfactor planar algebras (the more usual sense).

The most direct construction of the $T_n$ planar algebra is to interpret the single strand as $f^{(2n-2)}$ in the Temperley-Lieb planar algebra $A_{2n}$,
allowing arbitrary Temperley-Lieb diagrams with $(2n-2) m$ boundary points in the $m$-boxes. (Another way to say this, in the langauge of tensor categories with a distinguished tensor generator,
is to take the even subcategory of $A_{2n}$, thought of as generated by $f^{(2n-2)}$.) This certainly ensures that $T_n$ exists; below
we give a presentation by generators and relations.

We consider a skein theory with a $(k=2n+1)$ strand generator (allowing
in this appendix boxes with odd numbers of boundary points), at the
special value $q=e^{\frac{i\pi}{k+2}}$, and relations analogous to those of Definition \ref{def:pa}:
\begin{enumerate}

\item\label{delta-T} a closed loop is equal to $2 \cos(\frac{\pi}{k+2})$,

\item\label{rotateS-T}
\inputtikz{rotateS}

\item\label{capS-T}
\inputtikz{capS-T}

\item\label{twoS-T}
\inputtikz{twoS-T}
\end{enumerate}

A calculation analogous to that of Theorem \ref{thm:passacrossS} shows that we have the relations
\begin{equation*}
\inputtikz{pullstringoverS-T}
\qquad \text{and} \qquad \inputtikz{pullstringunderS-T}.
\end{equation*}
where $Z^- = Z^+ = (-1)^{\frac{k+1}{2}}$. (Recall that in the $D_{2n}$ case discussed in the body of the paper we had $Z^\pm = \pm 1$.)
These relations allow us to repeat the arguments showing that closed diagrams can be evaluated, and that the planar algebra is spherical.
When $k \equiv 3 \pmod{4}$ and $Z^\pm = +1$, the planar algebra $T_n$ is braided.

When $k \equiv 1 \pmod{4}$ and $Z^\pm = -1$, one can replace the usual crossing in Temperley-Lieb with minus itself; this is still a braiding on
Temperley-Lieb. One then has instead $Z^\pm = +1$, and so the entire planar algebra is then honestly braided.
Notice that $T_n$ is related to $A_{2n}$ in two \emph{different} ways. First, $T_n$ contains $A_{2n}$ as a subplanar algebra (simply because any planar algebra at a special value
of $\qi{2}$ contains the corresponding Temperley-Lieb planar algebra). Second, $T_n$ is actually the even part of $A_{2n}$, with an
unusual choice of generator (see above). The first gives a candidate braiding -- as we've seen it's only an `almost braiding' when $k \equiv 1 \pmod{4}$.
The second automatically gives an honest braiding, and in the $k \equiv 1 \pmod{4}$ case it's the negative of the first one.

Following through the consistency argument of \S \ref{sec:consistency}, mutatis mutandi,
we see that these relations do not collapse the planar algebra to zero. Further, along the lines of \S \ref{sec:category}, we can show that the
tensor category of projections is semisimple, with $\{f^{(0)}, f^{(1)}, \ldots, f^{(\frac{k-1}{2})}\}$ forming a complete orthogonal set of minimal projections.
The element $S$ in the planar algebra gives rise to isomorphisms $f^{(i)} \iso f^{(k-i)}$ for $i= 0, \ldots, \frac{k-1}{2}$. Further, the principal graph
is $T_{\frac{k-1}{2}}$, the tadpole graph:
\begin{equation*}
\mathfig{0.5}{graphs/Tk}.
\end{equation*}