Kauffman bracket skein modules of small 3-manifolds

The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed 3-manifolds are finitely generated over Q(A). In this paper, we develop a novel method for computing these skein modules. We show that if the skein module S(M,Q[A^±1]) of M is tame (e.g. finitely generated over Q[A^±1]), and the SL(2,C)-character scheme is reduced, then the dimension dim_Q(A)⁡S(M,Q(A)) is the number of closed points in this character scheme. This, in particular, verifies a conjecture in the literature relating dim_Q(A)⁡S(M,Q(A)) 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(A).