On the KBSM of links in lens spaces

TL;DR

This paper investigates the properties of the Kauffman bracket skein module of lens spaces, demonstrating its effectiveness as an invariant that can distinguish links with equivalent lifts in the 3-sphere.

Contribution

It introduces methods to compute the skein module for links in lens spaces and shows its effectiveness as an invariant that can differentiate links with identical lifts.

Findings

KBSM can distinguish inequivalent links with the same lift.

Transformation between band and disk diagrams facilitates computation.

KBSM relates to the Kauffman bracket of the lift in the 3-sphere.

Abstract

In this paper the properties of the Kauffman bracket skein module of $L(p,q)$ are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the Kauffman bracket skein module on an interesting class of examples consisting of inequivalent links with equivalent lifts in the $3$-sphere. The computation show that the Kauffman bracket skein module is an essential invariant, that is, it may take different values on links with equivalent lifts. We also show how the invariant is related to the Kauffman bracket of the lift in the $3$-sphere.