Subfactors of Index Less Than 5, Part 3: Quadruple Points
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 4valent vertices (in particular, the "weeds" $${\mathcal{Q}}$$ and $${\mathcal{Q}'}$$ from Part 1 (Morrison and Snyder in Commun. Math. Phys., doi:10.1007/s002200121426y, 2012). 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 $${\mathcal{Q}'}$$ immediately. The second uses connections, and when combined with an additional number theoretic argument it eliminates both weeds $${\mathcal{Q}}$$ and $${\mathcal{Q}'}$$ . Finally, we prove the uniqueness (up to taking duals) of the 3311 Goodmande la HarpeJones subfactor using a combination of planar algebra techniques and connections.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 December 2012
 DOI:
 10.1007/s0022001214725
 arXiv:
 arXiv:1109.3190
 Bibcode:
 2012CMaPh.316..531I
 Keywords:

 Fusion Category;
 Graph Automorphism;
 Gauge Class;
 Fusion Ring;
 Quadruple Point;
 Mathematics  Operator Algebras;
 Mathematics  Quantum Algebra
 EPrint:
 21 pages