Skein theory for the ADE planar algebras

TL;DR

This paper develops a skein theory framework for ADE planar algebras, providing generators, relations, a basis, and an algorithm for diagram simplification, advancing the understanding of subfactor planar algebras.

Contribution

It introduces a skein-theoretic approach to ADE planar algebras, including explicit generators, relations, and computational tools for diagram analysis.

Findings

Established generators and relations for ADE planar algebras

Provided a basis for these algebras

Developed an algorithm to express diagrams as linear combinations

Abstract

We give generators and relations for the planar algebras corresponding to $ADE$ subfactors. We also give a basis and an algorithm to express an arbitrary diagram as a linear combination of these basis diagrams.