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](http://arxiv.org/PS_cache/arxiv/pdf/0808/0808.0764v3.pdf), 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.



What are some natural and attractive open problems in Jones's theory of planar algebras?
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](http://arxiv.org/abs/0909.4099)). 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.


