The little desert? Some subfactors with index in the interval (5,3 +√5)

Progress on classifying small index subfactors has revealed an almost empty landscape. In this paper we give some evidence that this desert continues up to index $3+\sqrt{5}$. There are two known quantum-group subfactors with index in this interval, and we show that these subfactors are the only way to realize the corresponding principal graphs. One of these subfactors is 1-supertransitive, and we demonstrate that it is the only 1-supertransitive subfactor with index between 5 and $3+\sqrt{5}$. Computer evidence shows that any other subfactor in this interval would need to have rank at least 38. We prove our uniqueness results by showing that there is a unique flat connection on each graph. The result on 1-supertransitive subfactors is proved by an argument using intermediate subfactors, running the odometer from the FusionAtlas' Mathematica package and paying careful attention to dimensions. This is the published version of arXiv:1205.2742.