On the KBSM of links in lens spaces

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.