Skein Theory of Affine ADE Subfactor Planar Algebras

Abstract

The Kuperberg Program seeks to describe presentations of subfactor planar algebras in order to classify them and prove results about their corresponding categories purely diagrammatically. This program has been completed for index less than 4 and remains an area of ongoing research for index greater than 4. This thesis advances the program at index 4. At this index, planar algebras other than Temperley-Lieb have an affine A, D, or E principal graph. We give generators-and-relations presentations for all affine A subfactor planar algebras. Exclusively using the planar algebra language, we give new proofs determining the number of subfactors of the hyperfinite II₁ factor with principal graph affine A.

We then complete the Kuperberg Program for unshaded subfactor planar algebras of index 4 with principal graph affine A, D, and E₇. Categories corresponding to some of the affine A planar algebras are monoidally equivalent to cyclic pointed fusion categories. These categories are also monoidally equivalent to a representation category of a cyclic subgroup of SU(2). We give new, fully diagrammatic proofs of these two equivalences. In doing so, we yield novel diagrammatics for the representation and fusion categories.

Finally, to prove sufficiency of the affine D and E₇ presentations, we define jellyfish algorithms. In the affine D case, we present the first jellyfish algorithm implemented on a planar algebra with multiple generators. For affine E₇, we describe the jellyfish algorithm using a braiding and show directly that it is a well-defined surjective function onto C.