Dimension of the skein module of a Dehn filling
Edwin Kitaeff
TL;DR
The paper studies the dimension of the Kauffman bracket skein module after Dehn filling under a finiteness hypothesis for the exterior skein module, proving that for almost all slopes and almost all odd-order roots of unity, the complex dimension of the specialized skein matches the rational dimension of the ordinary skein. It develops a finitely generated localization $S(E_K(r),R_U)$ using the Frohman–Gelca basis and translates Dehn-surgery relations into module-theoretic constraints, yielding dimension equality after localization and specialization. When the character variety is finite, it further decomposes $S_\zeta(M)$ into localized skein modules at characters, with non-central contributions counted by their multiplicities; this relies on almost Azumaya properties and TehFroKan-type localization theory. The results extend previous tameness conditions and connect skein theory with character varieties and Azumaya loci, providing computable, principled dimension data for skein modules of Dehn fillings.
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 Lê, this condition is satisfied for $K$ a two-bridge knot. In this setting, we show that $\dim_{\mathbb{C}} (S_ζ(E_K (r))) = \dim_{\mathbb{Q} (A)} (S (E_K (r)))$ for almost all primitive roots of unity $ζ$ 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 $S_ζ(M)$ 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.