Subfactors of index less than 5, part 3: quadruple points

TL;DR

This paper develops new quadruple point obstructions to classify small index subfactors, successfully eliminating certain candidate graphs and establishing the uniqueness of a specific subfactor using advanced algebraic techniques.

Contribution

It introduces two novel quadruple point obstructions and proves the uniqueness of the 3311 subfactor, advancing the classification of subfactors with index less than 5.

Findings

Eliminates candidate graphs Q and Q' using quadruple point obstructions.

Proves the uniqueness of the 3311 subfactor.

Develops new techniques combining quadratic tangles and connections.

Abstract

One major obstacle in extending the classification of small index subfactors beyond 3+\sqrt{3} is the appearance of infinite families of candidate principal graphs with 4-valent vertices (in particular, the "weeds" Q and Q' from Part 1 (arXiv:1007.1730)). Thus instead of using triple point obstructions to eliminate candidate graphs, we need to develop new quadruple point obstructions. In this paper we prove two quadruple point obstructions. The first uses quadratic tangles techniques and eliminates the weed Q' immediately. The second uses connections, and when combined with an additional number theoretic argument it eliminates both weeds Q and Q'. Finally, we prove the uniqueness (up to taking duals) of the 3311 Goodman-de la Harpe-Jones subfactor using a combination of planar algebra techniques and connections.