 
Summary: SKEIN THEORY FOR THE ADE PLANAR ALGEBRAS
STEPHEN BIGELOW
Abstract. We give generators and relations for the planar algebras corre
sponding to the E8 subfactor. We also give a set of diagrams that forms a
basis, and an algorithm to express an arbitrary diagram as a linear combi
nation of basis diagrams. We summarize the analogous results for the other
ADE subfactors, which are obtained by the same methods.
1. Introduction
The notion of a planar algebra is due to Jones [Jon99]. The roughly equivalent
notion of a spider is due to Kuperberg [Kup96]. Planar algebras arise in many
contexts where there is a reasonably nice category with tensor products and duals.
Examples are the category of representations of a quantum group, or of bimodules
coming from a subfactor.
The subfactor planar algebras of index less than 4 can be classified into the two
infinite families AN and D2N , and the two sporadic examples E6 and E8. See
[GHJ89], [Ocn88], [Izu91], and [KO02] for the story of this "ADE" classification.
The Kuperberg program can be summarized as follows.
Give a presentation for every interesting planar algebra, and prove
as much as possible about the planar algebra using only its presen
tation.
