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

In this series of papers we show that there are exactly ten subfactors, other than $A_{\mathrm{\infty }}$ 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.