An effective proof of finiteness for Kauffman bracket skein modules

TL;DR

This paper proves a finiteness conjecture for Kauffman bracket skein modules of 3-manifolds with boundary, providing an alternative proof for closed manifolds and confirming the non-emptiness of peripheral ideals.

Contribution

It offers a constructive proof of the finiteness conjecture for skein modules, extending results to manifolds with boundary and addressing an open question about peripheral ideals.

Findings

Finiteness of Kauffman bracket skein modules for 3-manifolds with boundary

Alternative proof of Witten's finiteness conjecture for closed 3-manifolds

Peripheral ideal of any link is non-empty

Abstract

We prove a version of the finiteness conjecture for Kauffman bracket skein modules of $3$-manifolds with boundary, which was introduced by the second author in \cite{Det21}. In particular our methods, which are constructive, give an alternative proof of Witten's finiteness conjecture for the Kauffman bracket skein modules of closed $3$-manifolds, which was originally proved in \cite{GJS19}. Moreover, as a corollary we show that the peripheral ideal of any link is non-empty, answering a question of Frohman, Gelca and Lofaro \cite{FGL02}.