Rings of $S{L}_2(\mathbb{C})$-Characters and the Kauffman Bracket Skein Module

Abstract

Let $M$ be a compact orientable 3-manifold. The set of characters of $S{L}_2(\mathbb{C})$ 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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