Dimension of the skein module of a Dehn filling

Abstract

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 $S(\partial E_K ,\mathbb{Q} [A^{\pm 1}])$. As shown in a paper of Thang L\^e, 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.