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 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.

21 pages

References in corpus (7)

• Exotic subfactors of finite depth with Jones indices (5+sqrt{13})/2 and (5+sqrt{17})/2
• Monoidal 2-structure of Bimodule Categories
• The Annular Structure of Subfactors
• Cyclotomic integers, fusion categories, and subfactors
• Subfactors of index less than 5, part 2: triple points
• Subfactors of index less than 5, part 1: the principal graph odometer
• Subfactors of index less than 5, part 4: vines

Cited by in corpus (15)

• The classification of subfactors of index at most 5
• Exchange relation planar algebras of small rank
• Composed inclusions of $A_3$ and $A_4$ subfactors
• Subfactors of index less than 5, part 1: the principal graph odometer
• 1-supertransitive subfactors with index at most 6+1/5
• A planar calculus for infinite index subfactors
• 2-supertransitive subfactors at index $3+\sqrt{5}$
• The little desert? Some subfactors with index in the interval (5,3+\sqrt{5})
• The centre of the extended Haagerup subfactor has 22 simple objects
• Subfactors of index exactly 5
• Singly generated planar algebras of small dimension, Part III
• An obstruction to subfactor principal graphs from the graph planar algebra embedding theorem
• A rotational approach to triple point obstructions
• Braided Subfactors, Spectral Measures, Planar algebras and Calabi-Yau algebras associated to SU(3) modular invariants
• On the magnitudes of some small cyclotomic integers