  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.
