Infinite families of hyperbolic $3$-manifolds with finite dimensional skein modules

Abstract

The Kauffman bracket skein module $K(M)$ of a $3$-manifold $M$ is the quotient of the $\mathbb{Q}(A)$-vector space spanned by isotopy classes of links in $M$ by the Kauffman relations. A conjecture of Witten states that if $M$ is closed then $K(M)$ is finite dimensional. We introduce a version of this conjecture for manifolds with boundary and prove a stability property for generic Dehn-filling of knots. As a result we provide the first hyperbolic examples of the conjecture, proving that almost all Dehn-fillings of any two-bridge knot satisfies the conjecture.

References in corpus (7)

• Skein modules of 3-manifolds
• Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations
• On the Kauffman bracket skein module of the quaternionic manifold
• Character varieties, A-polynomials, and the AJ Conjecture
• 9 generators of the skein space of the 3-torus
• KBSM of the product of a disk with two holes and S^{1}
• On the Kauffman bracket skein module of surgery on a (2,2b) torus link

Cited by in corpus (3)

• Kauffman bracket skein modules of small 3-manifolds
• A basis for the Kauffman skein module of the product of a surface and a circle
• Generating sets for the Kauffman skein module of a family of Seifert fibered spaces