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

TL;DR

This paper classifies all subfactors with index between 4 and 5, extending previous classifications up to index 3+√3, and introduces a framework for analyzing these subfactors via principal graph odometers.

Contribution

It provides a complete classification of subfactors with index less than 5, identifying ten new subfactors beyond the known A_infinity family, and develops a novel approach using principal graph odometers.

Findings

Identified exactly ten subfactors with index between 4 and 5.

Extended classification from index 3+√3 to less than 5.

Introduced principal graph odometer method for analysis.

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.