I'm hoping to learn something about planar algebras while attacking a planar algebra question with an undergrad research student. I'm thinking about reading this paper, as Kuperberg's program seems like the sort of thing I'm looking for, but maybe there are better ideas:
What are some open questions in planar algebra theory that are self-contained within, and need minimal motivation outside, the planar algebra formalism?
If there are such questions that are well-known in the community, I'd appreciate being pointed to some of them. Short answers and references would be fine.
Some supplemental questions that would also be helpful, if answered:
What are the main open problems in the theory of planar algebras, proper? (This is kind of silly, because these are certainly the ones corresponding to the important open questions in subfactor theory. Perhaps, though, an answer to a question in planar algebras would resolve deep things in several areas where planar algebras appear. Such a question I'd consider a question in planar algebras, proper.)
Which of the main problems driving this subject are interesting even if they are not directly traced back to their implications in subfactor theory?
Of course, the theory of subfactors is the clear motivation for using the formalism. However, the diverse examples of planar algebras give evidence that we should study them in their own right.
Ideally, I am looking for problems that can be stated in the planar algebra formalism and do not require strong, direct reference back to the "subfactor world" to motivate.