Kauffman bracket skein modules of small 3-manifolds

Abstract

The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed 3-manifolds are finitely generated over $Q\left(A\right)$. In this paper, we develop a novel method for computing these skein modules.

We show that if the skein module $S(M,Q[A^{\pm 1}\left]\right)$ of M is tame (e.g. finitely generated over $Q\left[A^{\pm 1}\right]$), and the $SL(2,C)$-character scheme is reduced, then the dimension ${\dim }_{Q\left(A\right)}S(M,Q(A\left)\right)$ is the number of closed points in this character scheme. This, in particular, verifies a conjecture in the literature relating ${\dim }_{Q\left(A\right)}S(M,Q(A\left)\right)$ to the Abouzaid-Manolescu $SL(2,C)$-Floer theoretic invariants, for infinite families of 3-manifolds.

We prove a criterion for reducedness of character varieties of closed 3-manifolds and use it to compute the skein modules of Dehn fillings of $(2,2n+1)$-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds.

We also prove that the skein modules of rational homology spheres have dimension at least 1 over $Q\left(A\right)$.