Constructing the extended Haagerup planar algebra

TL;DR

This paper constructs a new subfactor planar algebra with the extended Haagerup principal graph, completing the classification of certain subfactors, and provides explicit descriptions and evaluation algorithms for it.

Contribution

It introduces a new subfactor planar algebra with the extended Haagerup graph, proving its uniqueness and providing explicit skein theoretic and algebraic descriptions.

Findings

Constructed a new subfactor planar algebra with extended Haagerup graph

Proved the uniqueness of this subfactor planar algebra

Developed an explicit evaluation algorithm for diagrams

Abstract

We construct a new subfactor planar algebra, and as a corollary a new subfactor, with the `extended Haagerup' principal graph pair. This completes the classification of irreducible amenable subfactors with index in the range $(4,3+\sqrt{3})$, which was initiated by Haagerup in 1993. We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. In the skein theoretic description there is an explicit algorithm for evaluating closed diagrams. This evaluation algorithm is unusual because intermediate steps may increase the number of generators in a diagram.