Computation of Lickorish's three manifold invariant using Chern-Simons theory

It is well known that any three-manifold can be obtained by surgery on a fr amed link in S-3. Lickorish gave an elementary proof for the existence of t he three-manifold invariant of Witten using a framed link description of th e manifold and the formalisation of the bracket polynomial as the Temperley -Lieb Algebra. Kaul determined a three-manifold invariant from link polynom ials in SU(2) Chern-Simons theory. Lickorish's formula for the invariant in volves computation of bracket polynomials of several cables of the link. We describe an easier way of obtaining the bracket polynomial of a cable usin g representation theory of composite braiding in SU(2) Chern-Simons theory. We prove that the cabling corresponds to taking tensor products of fundame ntal representations of SU(2). This enables us to verify that the two appar ently distinct three-manifold invariants are equivalent for a specific rela tion of the polynomial variables.