Kauffman bracket skein modules of small 3-manifolds

TL;DR

This paper introduces a new method for computing Kauffman bracket skein modules of small 3-manifolds, linking their dimension to character varieties and providing explicit calculations for certain hyperbolic manifolds.

Contribution

It develops a novel computational approach for skein modules, relates their dimension to character varieties, and computes skein modules for specific hyperbolic 3-manifolds.

Findings

Dimension of skein modules equals the number of points in the character variety.

Computed skein modules for Dehn fillings of certain torus knots and the figure-eight knot.

Established skein modules of rational homology spheres have dimension at least one.

Abstract

The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $3$-manifolds are finitely generated over $\mathbb Q(A)$. In this paper, we develop a novel method for computing these skein modules. We show that if the skein module $S(M,\mathbb Q[A^{\pm 1}])$ of $M$ is tame (e.g. finitely generated over $\mathbb Q[A^{\pm 1}]$), and the $SL(2,\mathbb C)$-character variety is reduced, then the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ is the number of closed points in this character variety. This, in particular, verifies a conjecture in the literature that relates the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ to the Abouzaid-Manolescu $SL(2,\mathbb C)$-Floer theoretic invariants, for large families of 3-manifolds. We also prove a criterion for reduceness 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 $\mathbb Q(A)$.