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?



What is the subfactor planar algebra of type $\tilde{A}_n$, of index 4?
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.


