Dimension of the skein module of a Dehn filling

TL;DR

This paper investigates the dimension of the Kauffman bracket skein module of Dehn filled 3-manifolds, establishing conditions under which the dimension at roots of unity matches the generic case and exploring decompositions related to character varieties.

Contribution

It demonstrates that the skein module dimension at roots of unity equals the generic dimension for almost all slopes and roots, extending understanding of skein modules in Dehn fillings.

Findings

Dimension equality at roots of unity for almost all slopes and roots

Finite character variety implies skein module decomposition

Localized skein modules' dimension equals multiplicity of points

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.