What is the subfactor planar algebra of type $\tilde{A}_n$, of index 4? As I understand it, there is a subfactor whose principal graph is the affine Dynkin diagram $\tilde{A}_n$. Since every vertex has two neighbors, does that mean the space of 1-boxes is two dimensional? Is that allowed?
 A: Yes, it is two dimensional, and this is allowed. It just means the planar algebra is not irreducible. I don't know of anyone that has thought about a presentation by generators and relations of this planar algebra yet. 
One issue here is that since $d=[M\colon N]^{1/2}=2$ is not generic ($>2$), one has to be careful about the annular multiplicities of the subfactor (arXiv:math/0105071). So I don't know if the planar algebra qualifies as "annular multiplicities $*10$" like (extended) Haagerup.
A: One concrete realization of this planar algebra goes as follows.  


*

*Start with the (generalized) PA freely generated by an oriented strand.

*Impose "$Z$-homology" relations: (a) oriented saddle moves, and (b) erase small loops (loop value = 1).

*Now introduce an unoriented strand which is the formal direct sum of an upward pointing strand and a downward pointing strand.

*If we now draw the principal graph of this PA from the point of view of the newly introduced unoriented strand type, we get the $\tilde{A}_\infty$ graph.

*I just noticed that you want $\tilde{A}_n$, not $\tilde{A}_\infty$.  For this case, start with $Z/n$ homology instead of $Z$ homology.  If you want more details let me know.

