Skein modules and character varieties of Seifert manifolds

We show that the Kauffman bracket skein module of a closed Seifert fibered 3-manifold $M$ is finitely generated over $\mathbb{Z}[A^{\pm 1}]$ if and only if $M$ is irreducible and non-Haken. We analyze in detail the character varieties $X(M)$ of such manifolds and show that under mild conditions they are reduced. We compute the Kauffman bracket skein modules for these $3$-manifolds (over $\mathbb{Q}(A)$) and show that their dimensions coincide with $\mathrm{\mid }X(M)\mathrm{\mid }\mathrm{.}$