Classification of spin models for Yang-Baxter planar algebras

Abstract

In this paper, we classify all spin models for singly-generated Yang-Baxter planar algebras in terms of certain highly regular graphs. Using Liu's classification of singly generated Yang-Baxter planar algebras, this classifies all spin models for the Jones polynomial, the Bisch-Jones planar algebras, and the Kauffman polynomial. This simplifies and clarifies Jaeger's classification of spin models for the Kauffman polynomial. In particular, we completely avoid the opaque and computationally-taxing concept of "formal self-duality"; we also explain the numerous exceptional cases in Jaeger's classification by demonstrating a new, discrete two-parameter family of spin models for the Bisch-Jones planar algebras.