Skein theory of planar algebras and some applications

Abstract

Modern subfactor theory was initiated by Jones by proving the significant result, Jones' index theorem. Later, Jones gave the topological axiomatization of the standard invariant of subfactors. This topological perspective suggests us to study subfactors via their skein theory, which is analogous to the presentation theory for groups via generators and relations. Therefore, one can ask for a classification of subfactor planar algebras via skein theory. In this paper, we give an example of One-Way Yang-Baxter planar algebras and provide a skein theory as a follow-up of the classification of Yang-Baxter planar algebras. We give a complete classification of 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. With the help of the planar algebra viewpoint, the connection between subfactor theory and geometric group theory was made through Jones’ construction of representations of the Thompson group F and T. Interesting subgroups of the Thompson group are constructed by considering the stabilizer of some vectors. We show that the topological relations from the planar algebras are sufficient to determine the algebraic structure. In particular, we determine the Jones subgroup and the 3-colorable subgroup uniformly.