Skein theory for subfactor planar algebras

Abstract

We find a subfactor planar algebra whose generating funtion is undetermined and another subfactor planar algebra which is not finitely generated. We also give a complete classification of singly generated Yang-Baxter relation planar algebras which leads to the discovery of a new one parameter family of planar algebras. At roots of unity, we obtain a sequence of subfactor planar algebras. While constructing these subfactor planar algebras, we overcome the three fundamental problems for skein theory: Evaluation, Consistency and Positivity. We also construct some other subfactor planar algebras and fusion categories from this family. In particular, another sequence of subfactor planar algebras is obtained which is an extension of the near group subfactor for $mathbb{Z}_4$. Two of these families of fusion categories can be thought of as the representation category of exceptional subgroups of quantum $SU(N)$ at level $Npm2$.