Subfactors of index less than 5, part 2: triple points

TL;DR

This paper reviews obstructions to classifying subfactors with principal graphs starting with a triple point, applying advanced techniques to eliminate certain candidate graphs for subfactors with index below 5.

Contribution

It introduces a novel application of Jones's quadratic tangles and connections techniques to classify subfactors with index less than 5, eliminating several candidate graphs.

Findings

Eliminated three of five candidate principal graph families.

Applied Jones's quadratic tangles in a new way.

Used connections techniques to restrict subfactor classifications.

Abstract

We summarize the known obstructions to subfactors with principal graphs which begin with a triple point. One is based on Jones's quadratic tangles techniques, although we apply it in a novel way. The other two are based on connections techniques; one due to Ocneanu, and the other previously unpublished, although likely known to Haagerup. We then apply these obstructions to the classification of subfactors with index below 5. In particular, we eliminate three of the five families of possible principal graphs called "weeds" in the classification from arXiv:1007.1730.