The classification of low index, finite-depth subfactor planar algebras seems to be a difficult problem. Below index 4, there’s a wonderful ADE classification. The type A planar algebras are just Temperley-Lieb at various roots of unity (and so the same as , as long as you change the pivotal structure). The type D planar algebras (with principal graphs the Dynkin diagrams ) were the subject of Noah’s talk at the conference, and the and planar algebras are nicely described in Stephen Bigelow’s recent paper.
But what happens as we go above index 4? In 1994 Haagerup gave a partial classification up to index . He showed that the only possible principal graphs come in two infinite families
(in both cases here the initial arm increases in steps of length 4) and another possibility
This result really opened a can of worms. Which of these graphs are actually realised? (Hint, they’re nicely colour-coded :-) What about higher index? What does it all mean? Are these graphs part of some quantum analogue of the classification of finite simple groups? Read one for the answer to the first question, at least.
A few years after this classification was announced (it’s actually never been proved in print; Haagerup’s paper only goes up to index ), Haagerup and Asaeda constructed the first graph in the first family (where “constructing a graph” means “constructing a subfactor or subfactor planar algebra whose principal graph is the graph”, of course), and the last graph. These graphs are now commonly referred to as the Haagerup graph, and the Asaeda-Haagerup graph. Bisch ruled out all the hexagon graphs, and Asaeda and Yasuda recently ruled out all the graphs in the first family except the first two!
This leaves just one, mysterious graph,
called the “extended Haagerup” graph, and no one has known whether it really comes from a subfactor planar algebra or not. Happily, the mystery is now solved, as we (Stephen Bigelow, Emily Peters, Noah Snyder and me) recently found a construction for it. There’s no scary subfactor nonsense, lots of pretty pictures, and even some jellyfish.
We give a “generators mod relations” presentation of the planar algebra, so it’s essentially pure skein theory, and very accessible. We need a few tricks however. Proving that a set of skein theory relations are consistent (that is, they don’t collapse the planar algebra to zero) is in general very hard, and we get around this problem by embedding our planar algebra inside the “graph planar algebra”. This is exactly analogous to proving that a group described by generators and relations isn’t trivial by embedding it into a symmetric group. Proving that you can evaluate every closed diagram in our planar algebra requires some nifty skein theory, shown in the last few slides.
Our construction relies heavily on Jones’ techniques for analysing the representation theory of the annular Temperley-Lieb category, and follows almost identically Emily Peters’ recent re-construction of the Haagerup subfactor. If you want to understand more detail beyond what’s in the slides, you should definitely read her paper. The differences between the two cases are that to construct extended Haagerup, you need to solve a huge family of overdetermined quadratics in a 148475-dimensional vector space, while for Haagerup the corresponding space was only 375-dimensional, and that the skein theory requires a bit more work.