The Kauffman bracket skein module at an irreducible representation

Abstract

In this paper, we study the Kauffman bracket skein module of closed oriented three-manifolds at non-multiple-of-four roots of unity. Our main result establishes that the localization of these modules at a maximal ideal, which corresponds to an irreducible representation of the fundamental group of the manifold, forms a one-dimensional free module over the localized unreduced coordinate ring of the character variety. We apply this to show that the dimension of the skein module of an oriented rational homology sphere with finite character variety and at most $2$-torsion in its first homology is greater than or equal to the dimension of the unreduced coordinate ring of the character variety. This leads to a computation of the dimension of the skein module with coefficients in rational functions for such rational homology spheres with tame universal skein module.