KBSM of the product of a disk with two holes and S^{1}

TL;DR

This paper develops diagrammatic techniques and Reidemeister moves for links in surfaces times S^{1}, enabling new computations of Kauffman Bracket Skein Modules for specific 3-manifolds involving a disk with two holes and S^{1}.

Contribution

It introduces a novel diagrammatic approach and computes KBSM for manifolds involving a disk with two holes and S^{1}, showing the module is free.

Findings

Computed KBSM for D^{2}xS^{1} and AxS^{1}

Derived KBSM for F_{0,3}xS^{1} and proved it is free

Presented new diagrammatic methods for links in surface x S^{1}

Abstract

We introduce diagrams and Reidemeister moves for links in FxS^{1}, where F is an orientable surface. Using these diagrams we compute (in a new way) the Kauffman Bracket Skein Modules (KBSM) for D^{2}xS^{1} and AxS^{1}, where D^{2} is a disk and A is an annulus. Moreover, we also find the KBSM for the F_{0,3}xS^{1}, where F_{0,3} denotes a disk with two holes, and thus show that the module is free.