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\begin{abstract}
We give a combinatorial description of the ``$D_{2n}$ planar algebra,''
by generators and relations. We explain how the generator
interacts with the Temperley-Lieb braiding.
This shows the previously known braiding on the even part
extends to a `braiding up to sign' on the entire planar algebra.

We give a direct proof that our relations are consistent (using this
`braiding up to sign'), give a complete description of the associated
tensor category and principal graph, and show that the planar algebra
is positive definite. These facts allow us to identify our combinatorial
construction with the standard invariant of the subfactor $D_{2n}$.
%Here, for the first time \textbf{[drumroll]}, the $D_{2n}$ planar algebra is presented in an entirely combinatorial fashion.
%\textbf{[fireworks]}  Subfactors are shunned and pictures abound. \textbf{[cymbal-clash]}
\end{abstract}

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\section{Introduction}\label{sec:intro}
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\section{Background}
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\section{Skein theory}
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\section{The planar algebra $\pa$ is $D_{2n}$}
This planar algebra is called $D_{2n}$ because it is the unique subfactor
planar algebra with principal graph $D_{2n}$.  To prove this we will need
two key facts: first that its principal graph is $D_{2n}$, and second that
it is a subfactor planar algebra.

In \S \ref{sec:category}, we describe a tensor category associated to any
planar algebra, and using that define the principal graph. We then check
that the principal graph for $\pa$ is indeed the Dynkin
diagram $D_{2n}$.

In \S \ref{sec:basis}, we exhibit an explicit basis for the planar
algebra.  This makes checking positivity straightforward.

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This paper is available online at \arxiv{0808.0764}, and at
\url{http://tqft.net/d2n}.

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