Chirality and principal graph obstructions

TL;DR

This paper introduces new obstructions for bipartite graphs to be principal graphs of subfactors, significantly advancing the classification by eliminating many potential graphs with initial triple and quadruple points.

Contribution

It develops generalized triple and quadruple point obstructions using planar algebra techniques, extending previous results and aiding subfactor classification.

Findings

Established a triple point obstruction generalizing all known initial obstructions.

Proved a quadruple point obstruction using similar techniques.

Eliminated infinite families of potential principal graphs with initial triple and quadruple points.

Abstract

Determining which bipartite graphs can be principal graphs of subfactors is an important and difficult question in subfactor theory. Using only planar algebra techniques, we prove a triple point obstruction which generalizes all known initial triple point obstructions to possible principal graphs. We also prove a similar quadruple point obstruction with the same technique. Using our obstructions, we eliminate some infinite families of possible principal graphs with initial triple and quadruple points which were a major hurdle in extending subfactor classification results above index 5.