On a presentation of the spin planar algebra

Abstract

We define a certain abstract planar algebra by generators and relations, study various aspects of its structure, and then identify it with Jones' spin planar algebra.Our goal in this note is to exhibit a presentation -a skein theory -for a very simple planar algebra, the spin planar algebra, which is well known from the very first paper [Jns1999] of Jones on planar algebras. Our technique is to define a certain abstract planar algebra by generators and relations, carefully study various aspects of its structure -including explicit bases for its vector spaces -and then identify it with Jones' spin planar algebra.We will assume throughout that the reader is familiar with planar algebras. Planar algebras are collections of vector spaces with an action by the operad of planar tangles. However, since the notion of planar algebras has been evolving since its definition in [Jns1999], to fix notations and definitions for the version of planar algebras that we use here, we refer to [DeKdy2018]. In particular, we use the version where the vector spaces are indexed by (k, ǫ) with k ∈ {0, 1, 2, · · · } and ǫ ∈ {±} as opposed to the older version. The equivalence between these two is also shown in [DeKdy2018]. Other notions such as universal planar algebras are treated carefully in [KdySnd2004] (in the context of the older planar algebras) and together these 3 papers cover all of the notions that are used here.Let S = {s 1 , s 2 , · · · , s n } be a finite set. We will define a planar algebra over C associated to this set. Begin with the label set L = L (0,−) = S equipped with the identity involution * . Consider the quotient P = P (L, R) of the universal planar algebra P (L) by the set R of relations in Figures 1 and 2 (where δ ij denotes the Kronecker delta). PSfrag replacements v + s i Institute,