Skein spaces and spin structures

Abstract

This paper relates skein spaces based on the Kauffman bracket and spin structures. A spin structure on an oriented 3-manifold provides an isomorphism between the skein space for parameter A and the skein space for parameter -A. There is an application to Penrose's binor calculus, which is related to the tensor calculus of representations of SU(2). The perspective developed here is that this tensor calculus is actually a calculus of spinors on the plane, and the matrices a re determined by a type of spinor transport which generalises to links in any 3-manifold. A second application shows that there is a skein space which is the algebra of functions on the set of spin structures for the 3-manifold.