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 1boxes is two dimensional? Is that allowed?
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.

1$\begingroup$ Kevin, can you tell me more about what $Z$homology and $Z/n$ homology mean here? $\endgroup$ May 24 '12 at 19:09

$\begingroup$ Z/n homology is a bit of an abbreviation, perhaps... If you're working on a closed surface, Z/n homology just means you can take n parallel (considering orientations, too) circles and remove then (even if they are essential on the surface). If you want to work on open surfaces (e.g. disks, to define a planar algebra), I think you need to add two nvalent vertices, one with all edges oriented in, the other with all oriented out. Now you can replace n parallel edges and replace them with a pair of these. ... $\endgroup$ May 25 '12 at 5:46

$\begingroup$ If you do the dimension count, each disk space is 1 or 0 dimensional, depending on whether the signed count of the boundary points is 0 mod n or not. It seems this should give $\widehat{A_n}$ graphs per Kevin's prescription above (although maybe not with exactly the same n?). $\endgroup$ May 25 '12 at 5:50

$\begingroup$ I suspect one can also freely chose the rotational eigenvalue of the two nvalent vertices, and these give the different nonisomorphic subfactor planar algebras with principal graph $\widehat{A_n}$. $\endgroup$ May 25 '12 at 5:50

$\begingroup$ Finally (to record all the thoughts from talking to Dave Penneys this afternoon), you ought to get the affine D subfactors via the automorphism in Kevin's picture which reverses all orientations. $\endgroup$ May 25 '12 at 5:51
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.