SUBFACTORS OF INDEX LESS THAN 5, PART 4: VINES

Abstract

We eliminate 39 infinite families of possible principal graphs as part of the classification of subfactors up to index 5. A number-theoretic result of Calegari–Morrison–Snyder, generalizing Asaeda–Yasuda, reduces each infinite family to a finite number of cases. We provide algorithms for computing the effective constants that are required for this result, and we obtain 28 possible principal graphs. The Ostrik d-number test and an algebraic integer test reduce this list to seven graphs in the index range (4,5) which actually occur as principal graphs.