Calculating two-strand jellyfish relations

David Penneys and Emily Peters

Vol. 277 (2015), No. 2, 463–510

DOI: 10.2140/pjm.2015.277.463

Abstract

We construct a $3^{\mathbb{Z}∕4}$ subfactor using an algorithm which, given generators in a spoke graph planar algebra, computes two-strand jellyfish relations. This subfactor was known to Izumi, but has not previously appeared in the literature. We systematically analyze the space of second annular consequences, adapting 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 and Penneys explains the connection between spoke subfactor planar algebras and the jellyfish algorithm, and work of Morrison and Penneys automates the construction of subfactors where both principal graphs are spoke graphs using one-strand jellyfish. This is the published version of arXiv:1308.5197.

Keywords

subfactors, planar algebras, jellyfish algorithm, principal graphs

Mathematical Subject Classification 2010

Primary: 46L37

Secondary: 18D05, 57M20

Milestones

Received: 3 June 2014

Revised: 9 January 2015

Accepted: 22 January 2015

Published: 15 September 2015