Dimension of the skein module of a Dehn filling

Given a knot $K$ and a generic slope $r$, we study the Kauffman bracket skein module (KBSM) $S(E_K(r),\mathbb{Q}(A))$ of the Dehn filling $E_K(r)$ of slope $r$ along $K$, assuming that the KBSM $S(E_K,\mathbb{Q}[A^{\pm 1}])$ of the exterior $E_K$ of $K$ is finitely generated over $S(\mathrm{\partial }{E}_K,\mathbb{Q}[A^{\pm 1}])$. As shown in a paper of Thang Lê, this condition is satisfied for $K$ a two-bridge knot. In this setting, we show that ${\dim }_{\mathbb{C}}(S_{\zeta }(E_K(r)))={\dim }_{\mathbb{Q}(A)}(S(E_K(r)))$ for almost all primitive roots of unity $\zeta$ of order $2N$ with $N$ odd, and for almost all slopes $r$. When the character variety of a 3-manifold $M$ is finite, we also discuss the decomposition of $S_{\zeta }(M)$ in terms of localized skein modules. In particular, the dimension of the localized skein modules at a non-central point is the multiplicity of this point.