Calculating two-strand jellyfish relations

TL;DR

This paper introduces an algorithm for constructing subfactors with spoke graph principal graphs using two-strand jellyfish relations, including a previously undocumented 3^{Z/4} subfactor, advancing the understanding of spoke subfactor planar algebras.

Contribution

It provides a systematic method for computing two-strand jellyfish relations and constructs new subfactors, expanding the known examples and theoretical framework.

Findings

Constructed a new 3^{Z/4} subfactor not previously documented.

Developed a systematic approach to second annular consequences.

Connected spoke subfactor planar algebras with the jellyfish algorithm.

Abstract

We construct subfactors where one of the principal graphs is a spoke graph using an algorithm which computes two-strand jellyfish relations. One of the subfactors we construct is a 3^{Z/4} subfactor known to Izumi, which has not previously appeared in the literature. To do so, we provide a systematic treatment of the space of second annular consequences, which is analogous to Jones' treatment of the space of first annular consequences in his quadratic tangles article. This article is the natural followup to two recent articles on spoke subfactor planar algebras and the jellyfish algorithm. Work of Bigelow-Penneys explains the connection between spoke subfactor planar algebras and the jellyfish algorithm, and work of Morrison-Penneys automates the construction of subfactors where both principal graphs are spoke graphs using one-strand jellyfish.