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.

9 pages, amstex, 15 figures. Revised by a substantial addition to give a geometrical description of all the commutative skein algebras, for A^6=1 (q^3=1)

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• The Localized Skein Algebra is Frobenius
• Double affine Hecke algebras and generalized Jones polynomials
• Two-dimensional state sum models and spin structures
• Finite presentations for stated skein algebras and lattice gauge field theory
• On framings of links in 3-manifolds
• Kauffman bracket skein modules of small 3-manifolds
• Character algebras of decorated SL_2(C)-local systems
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