Connections and planar algebras

With Emily Peters, I've been exploring subfactors with index in the interval $(5, 3+\sqrt{5})$. We've recently obtained a classification of 1-supertransitive subfactors in this range, and performed an extensive computer search in higher supertransitivities. I'll describe the examples of subfactors we've found. We have two main new techniques. First, even when we only know a fragment of a principal graph, we can extract certain inequalities by considering the norms of the entries of a connection. This allows the new classification result. Second, we extend the theory of bi-unitary connections to the bi-invertible case, and find we can then work over a fixed number field. This allows effective use of the "hybrid method" of constructing subfactors: given a not-necessarily flat bi-invertible connection, we can efficiently solve the equations for a flat element in the graph planar algebra. This lets us completely analyse the examples.