Classification of Thurston relation subfactor planar algebras

Abstract

Bisch and Jones proposed the classification of planar algebras by simple generators and relations. They investigated with the second author the classification of planar algebras generated by 2-boxes. In this paper, we classify singly-generated Thurston-relation planar algebras, defined as subfactor planar algebras generated by a 3-box satisfying a relation proposed by Dylan Thurston. Our main result shows that such subfactor planar algebras are either the $E_6$ subfactor planar algebras or belong to a two-parameter family of planar algebras arising from the representations of type $A$ quantum groups. We introduce a new method for determining positivity of the Markov trace of planar algebras in this family.