A rotational approach to triple point obstructions

TL;DR

This paper introduces a generalized triple point obstruction for subfactors, unifying existing obstructions using planar algebraic and connection-theoretic techniques centered around the rotation operator.

Contribution

It develops a new, more general triple point obstruction that encompasses previous obstructions, advancing the classification of small index subfactors.

Findings

Unified the triple-single and quadratic tangles obstructions

Provided new constraints for subfactors with 3-valent initial branching

Enhanced understanding of the role of the rotation operator in subfactor theory

Abstract

Subfactors where the initial branching point of the principal graph is 3-valent are subject to strong constraints called triple point obstructions. Since more complicated initial branches increase the index of the subfactor, triple point obstructions play a key role in the classification of small index subfactors. There are two strong triple point obstructions called the triple-single obstruction and the quadratic tangles obstruction. Although these obstructions are very closely related, neither is strictly stronger. In this paper we give a more general triple-point obstruction which subsumes both. The techniques are a mix of planar algebraic and connection-theoretic techniques with the key role played by the rotation operator.