Rings of $SL_2({\mathbb C})$-Characters and the Kauffman Bracket Skein Module

Details Rings of $SL_2({\mathbb C})$-Characters and the Kauffman Bracket Skein Module Rings of $SL_2({\mathbb C})$-Characters and the Kauffman Bracket Skein Module

1996-04-20

arxiv.org/abs/q-alg/9604014v1

Mathematics

Quantum Algebra

LaTeX2e v1.2, document class amsart.cls with package epsfig, 18 pages, 9 figures

Scientific paper

Let $M$ be a compact orientable 3-manifold. The set of characters of $SL_2({\mathbb C})$ representations of the fundamental group of $M$ forms a closed affine algebraic set. We show that its coordinate ring is isomorphic to a specialization of the Kauffman bracket skein module modulo its nilradical. This is accomplished by making the module into a combinatorial analog of the ring, in which tools of skein theory are exploited to illuminate relations among characters. We conclude with an application, proving that a small manifold's specialized module is necessarily finite dimensional.

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