2-supertransitive subfactors at index 3+5

We introduce a new method for showing that a planar algebra is evaluable. In fact, this method is universal for finite depth subfactor planar algebras. By making careful choices in the method's application, one can often significantly reduce the complexity of the computations.Using our technique, we prove existence and uniqueness of a subfactor planar algebra with principal graph consisting of a diamond with arms of length 2 at opposite sides, which we call “2D2”. This is expected to be the last remaining construction required for the classification of subfactor planar algebras up to index 3+5.This classification will also require showing the uniqueness of the subfactor planar algebra with principal graph 4442. We include a short proof of this fact, known to Izumi but as yet unpublished.This is the published version of arXiv:1406.3401.