Rings of SL₂(ℂ)-characters and the Kauffman bracket skein module

Let M be a compact orientable 3-manifold. The set of characters of SL₂(ℂ)-representations of π₁(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 realizing the module as 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.