Classification of Thurston-relation subfactor planar algebras

TL;DR

This paper classifies certain subfactor planar algebras generated by a 3-box satisfying Thurston's relation, identifying them as either $E^6$ or quantum $SU(N)$ representations, using new positivity and complexity reduction methods.

Contribution

It introduces a novel classification of 3-box generated subfactor planar algebras satisfying Thurston's relation, expanding understanding beyond 2-box cases.

Findings

Subfactor planar algebras are either $E^6$ or from quantum $SU(N)$ representations.

Developed new methods for positivity determination in planar algebras.

Created techniques to simplify complex computations in classification.

Abstract

Bisch and Jones suggested the skein theoretic classification of planar algebras and investigated the ones generated by 2-boxes with the second author. In this paper, we consider 3-box generators and classify subfactor planar algebras generated by a non-trivial 3-box satisfying a relation proposed by Thurston. The subfactor planar algebras in the classification are either $E^6$ or the ones from representations of quantum $SU(N)$. We introduce a new method to determine positivity of planar algebras and new techniques to reduce the complexity of computations.