2-supertransitive subfactors at index $3+\sqrt{5}$

TL;DR

This paper establishes the existence and uniqueness of specific subfactor planar algebras at index 3+√5, focusing on the 2D2 and 4442 principal graphs, advancing the classification of subfactors.

Contribution

It proves the existence and uniqueness of subfactor planar algebras with principal graphs 2D2 and 4442 at index 3+√5, filling gaps in subfactor classification.

Findings

Uniqueness of subfactor planar algebra with principal graph 2D2.

Uniqueness of subfactor planar algebra with principal graph 4442.

Conjecture that these results complete the classification at index 3+√5.

Abstract

This article proves the existence and uniqueness of a subfactor planar algebra with principal graph consisting of a diamond with arms of length 2 at opposite sides, which we call 2D2. We also prove the uniqueness of the subfactor planar algebra with principal graph 4442. We conjecture this will complete the list of subfactor planar algebras at index $3+\sqrt{5}$.