Subfactors of index less than 5, part 1: the principal graph odometer

Abstract

In this series of papers we show that there are exactly ten subfactors, other than $A_{\infty }$ subfactors, of index between 4 and 5. Previously this classification was known up to index $3+\sqrt{3}$. In the first paper we give an analogue of Haagerup's initial classification of subfactors of index less than $3+\sqrt{3}$, showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series.

36 pages (updated to reflect that the classification is now complete)

References in corpus (4)

• Cyclotomic integers, fusion categories, and subfactors
• Subfactors of index less than 5, part 2: triple points
• Subfactors of index less than 5, part 3: quadruple points
• On Haagerup's list of potential principal graphs of subfactors

Cited by in corpus (12)

• Subfactors of index less than 5, part 3: quadruple points
• Quadratic Tangles in Planar Algebras
• Exchange relation planar algebras of small rank
• Composed inclusions of $A_3$ and $A_4$ subfactors
• 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})
• Constructing spoke subfactors using the jellyfish algorithm
• A rotational approach to triple point obstructions
• An obstruction to subfactor principal graphs from the graph planar algebra embedding theorem
• Subfactors of index exactly 5