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.

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    $\begingroup$ @Jon: Are you familiar with The Secret Blogging Seminar, maintained by several people, including MO participants? It has a collection of articles on planar algebras: en.wordpress.com/tag/planar-algebras $\endgroup$ – Joseph O'Rourke Dec 10 '10 at 18:07
  • $\begingroup$ @Joseph: I saw this while looking through planar algebras tagged MO questions. This will be really helpful. Thanks for suggesting it. $\endgroup$ – Jon Bannon Dec 10 '10 at 18:33
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    $\begingroup$ I'd like to advertise mathoverflow.net/questions/35882/… part of which was asked by Vaughan Jones (see Noah Snyder's comment). I suppose the motivation is knot theory. One can make it more a "pure planar algebras" question by asking which planar algebras admit homomorphic expansions in the sense of Bar-Natan. I.e. which planar algebras give rise to finite-type invariants. As far as I know, this is wide open. $\endgroup$ – Daniel Moskovich Dec 10 '10 at 20:39
  • $\begingroup$ Thank you very much Daniel. This may be perfect for what I want to do. (I'm glad you decided to re-post it, because I was interested when I saw it appear earlier as an answer!) $\endgroup$ – Jon Bannon Dec 11 '10 at 0:36
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    $\begingroup$ This is a great question, and I actually think there are a lot of natural questions in the theory of planar algebras which aren't necessarily inspired by corresponding questions about subfactors (some come from knot theory, some from Lie theory). Just yesterday I was thinking it would be interesting to make such a list of open questions, but I probably won't have time to do a good job answering this for a few weeks. $\endgroup$ – Noah Snyder Jan 4 '11 at 3:08

One important kind of question in planar algebras is given generators and relations for a planar algebra can you:

  • Find an algorithm which takes an arbitrary closed diagram and evaluates it to give a number
  • Show that any two ways of evaluating a closed diagram gives the same answer
  • Find an explicit basis for every box space

Here's some examples. These are motivated by questions outside of planar algebras as such, but the motivations don't come from subfactor theory, and the questions are not hard to state in purely elementary terms.

  • Given the generators and relations defining the Yamada polynomial (this gives quantum so(3)) at circle value a real number 4 or bigger, can you find a manifestly positive evaluation algorithm? (This is the 4-color theorem.)
  • Write down the natural generators and relations for the planar algebra coming from the adjoint representation of Deligne's (conjectural!) exceptional Lie algebra. (These are antisymmetry, the Jacobi relation, and Vogel's relation for simplifying a square.) Answer each of the above three questions. This would give a construction of the exceptional Lie algebra.
  • Same as the last question but for the adjoint representation of Vogel's (conjectural) universal Lie algebra. Here you just have anti-symmetry, Jacobi, and killing negligibles.

Some other interesting questions not of the above type:

  • Find a continuous family of planar algebras which isn't one of: HOMFLY polynomial, Kauffman polynomial, the symmetric group S_t, Fuss-Catalan, or one of a few other known constructions.
  • Consider the planar algebra of planar graphs with a bipartite shadings of both their vertices and faces. Find explicit quotients of this planar algebra with finite dimensional box spaces. (Not counting the easy ones which come from relations simplifying 2-valent vertices.) This question sounds topological, but is secretly about quadrilaterals of factors.
  • Find a direct (i.e. non-number theoretic) proof that there exists a closed diagram which evaluates to two different numbers for each of the "further extended Haagerups" (see our paper with Bigelow-Morrison-Peters). Here the "jellyfish algorithm" gives an explicit way to evaluate closed diagrams (depending on some choices), and you just want to show that making different choices gives different answers.
  • $\begingroup$ I may be jumping the gun here, but I think this answer will do nicely. $\endgroup$ – Jon Bannon Apr 16 '11 at 19:57

I'd like to advertise Which presentations of (non)planar algebras give rise to knots? (see also Noah Snyder's comment there).
It is rare that one can provide a response to an MO question simply by linking to another.


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