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

## TL;DR

This paper proves that most Dehn-fillings of two-bridge knots result in hyperbolic 3-manifolds with finite dimensional Kauffman bracket skein modules, supporting a conjecture relating to quantum invariants of 3-manifolds.

## Contribution

It introduces a boundary version of Witten's conjecture and demonstrates its validity for a broad class of hyperbolic 3-manifolds obtained via Dehn-filling.

## Key findings

- Almost all Dehn-fillings of two-bridge knots satisfy the finite dimensional skein module conjecture.
- Established a stability property for skein modules under generic Dehn-filling.
- Provided the first hyperbolic examples supporting Witten's conjecture for closed 3-manifolds.

## 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.