Existence of the $AH+2$ subfactor

TL;DR

This paper proves the existence of the $AH+2$ subfactor, a specific self-dual subfactor with a precise index, using two different methods: direct graph connections and an indirect approach based on existing subfactor constructions.

Contribution

It provides two distinct proofs of the $AH+2$ subfactor's existence, enhancing understanding of its structure and relation to other subfactors.

Findings

Two proofs of the $AH+2$ subfactor existence.

Confirmation of the subfactor's index as (9+√17)/2.

Insights into the subfactor's structure and connections.

Abstract

We give two different proofs of the existence of the $AH+2$ subfactor, which is a $3$-supertransitive self-dual subfactor with index $\frac{9+\sqrt{17}}{2} $. The first proof is a direct construction using connections on graphs and intertwiner calculus for bimodule categories. The second proof is indirect, and deduces the existence of $AH+2$ from a recent alternative construction of the Asaeda-Haagerup subfactor and fusion combinatorics of the Brauer-Picard groupoid.