An Obstruction to Subfactor Principal Graphs from the Graph Planar Algebra Embedding Theorem

We find a new obstruction to the principal graphs of subfactors. It shows that in a certain family of 3-supertransitive principal graphs, there must be a cycle by depth 6, with one exception, the principal graph of the Haagerup subfactor. This is the published version of arXiv:1302.5148. A II1 subfactor is an inclusion A ⊂ B of infinite von Neumann algebras with trivial centre and a compatible trace with tr(1) = 1. In this setting, one can analyze the bimodules ⊗−generated by ABB and BBA. The principal graph of a subfactor has as vertices the simple bimodules appearing, and an edge between vertices X and Y for each copy of Y appearing inside X ⊗B. It turns out that this principal graph is a very useful invariant of the subfactor (at least in the amenable case), and many useful combinatorial constraints on graphs arising in this way have been discovered. As examples, see [MS12, Jon03, Sny12, Pen13]. Moreover, with sufficiently powerful combinatorial (and number theoretic, c.f. [Asa07, AY09, CMS11, Ost09]) constraints in hand, it has proved possible to enumerate all possible principal graphs for subfactors with small index. This approach was pioneered by Haagerup in [Haa94], and more recently continued, resulting in a classification of subfactors up to index 5 [MS12, MPPS12, IJMS12, PT12, JMS13]. In this note we demonstrate the following theorem, providing a combinatorial constraint on the principal graph of a subfactor, of a rather different nature than previous results. Theorem. If the principal graph of a 3-supertransitive II1 subfactor begins as