8. Subfactors

The supertransitivity with respect to an object $X$ with $\dim(X)>2$ is the largest $N$ such that $\text{Hom}(1,X^{\otimes n})$ is TemperleyLieb.
Supertransitivity is the analog of transitivity of group actions.Problem 8.1.
[Jones] Is there an upper bound on the supertransitivity of a subfactor planar algebra?
Remark. [org.aimpl.user:scott@tqft.net] Currently the extended Haagerup subfactor holds the record, with $n=7$. The AsaedaHaagerup subfactor has $n=5$, and otherwise all known examples have $n\leq 4$.


Problem 8.2.
[Snyder] Find a nonnumber theoretic argument to rule out the rest of the Haagerup family vine.
For example, is there a diagram that evaluates in two different ways? 
Problem 8.3.
[Morrison, Peters] Is there a polymer theory of principal graphs? What graphs can appear as subgraphs of principal graphs?
Penneys: Note that there are examples of forbidden subgraphs, e.g., part of the bad seed.
Cite this as: AimPL: Classifying fusion categories, available at http://aimpl.org/fusioncat.