Rings of SL2( $\mathbb{C}$ )-characters and the Kauffman bracket skein module

Abstract

Let M be a compact orientable 3-manifold. The set of characters of SL2($\mathbb{C}$)-representations of ${\pi }_1(M)$ 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.