Skein theory for the D_{2n} planar algebras

TL;DR

This paper provides a combinatorial framework for the D_{2n} planar algebra, detailing generators, relations, and braiding interactions, and confirms its positivity and correspondence with the subfactor's standard invariant.

Contribution

It introduces a new combinatorial description of the D_{2n} planar algebra with explicit generators, relations, and braiding properties, extending known structures.

Findings

The relations are consistent and define a positive definite planar algebra.

The braiding extends to a 'braiding up to sign' on the entire algebra.

The construction matches the standard invariant of the D_{2n} subfactor.

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 $D_{2n}$.