A Well-Defined Jellyfish Algorithm for the Affine $E_7$ Subfactor Planar Algebra

TL;DR

This paper provides a diagrammatic presentation of the affine E7 subfactor planar algebra and proves that its jellyfish algorithm is a well-defined invariant, advancing the understanding of subfactor theory.

Contribution

It introduces a new diagrammatic presentation for the affine E7 subfactor planar algebra and establishes the well-definedness of its jellyfish algorithm.

Findings

Jellyfish algorithm is a surjection onto complex numbers.

The algorithm is an invariant on closed diagrams.

Provides a new presentation for the affine E7 subfactor planar algebra.

Abstract

In this paper, we contribute to the Kuperberg program by giving a diagrammatic presentation of generators and relations for the affine $E_7$ unshaded subfactor planar algebra. Using this presentation, we prove that its jellyfish algorithm is a well-defined surjection onto $\mathbb{C}$. In particular, this shows that the jellyfish algorithm is an invariant on closed diagrams for this planar algebra.