Exchange relation planar algebras of small rank

TL;DR

This paper classifies exchange relation planar algebras with 4-dimensional 2-boxes, focusing on positivity, projection lattices, and biprojections, extending previous classifications of singly generated planar algebras.

Contribution

It introduces a classification of small-rank exchange relation planar algebras, emphasizing biprojection existence and extending prior work on singly generated cases.

Findings

Classified exchange relation planar algebras with 4-dimensional 2-boxes.

Established the existence of biprojections in these algebras.

Extended classification to certain families with specific 3-box and 2-box dimensional constraints.

Abstract

The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will discuss the lattice of projections of 2-boxes, specifically the rank of the projections. From this point, several results about biprojections are obtained. The key break of the classification is to show the existence of a biprojection. By this method, we also classify another two families of subfactor planar algebras, subfactor planar algebras generated by 2-boxes with 4 dimensional 2-boxes and at most 23 dimensional 3-boxes; subfactor planar algebras generated by 2-boxes, such that the quotient of 3-boxes by the basic construction ideal is abelian. They extend the classification of singly generated planar algebras obtained by Bisch, Jones and the author.