Skein spaces and spin structures

TL;DR

This paper explores the relationship between skein spaces derived from the Kauffman bracket and spin structures on 3-manifolds, revealing new connections with Penrose's binor calculus and spinor transport.

Contribution

It establishes an isomorphism between skein spaces at parameters A and -A using spin structures, and introduces a skein space representing functions on the set of spin structures.

Findings

Isomorphism between skein spaces for A and -A via spin structures

Connection of Penrose's binor calculus to tensor calculus of SU(2)

Existence of a skein space as the algebra of functions on 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.