  It is well known that any three-manifold can be obtained by surgery on a
framed link in $S^3$. Lickorish gave an elementary proof for the existence of
the three-manifold invariants of Witten using a framed link description of the
manifold and the formalisation of the bracket polynomial as the Temperley-Lieb
Algebra. Kaul determined three-manifold invariants from link polynomials in
SU(2) Chern-Simons theory. Lickorish's formula for the invariant involves
computation of bracket polynomials of several cables of the link. We describe
an easier way of obtaining the bracket polynomial of a cable using
representation theory of composite braiding in SU(2) Chern-Simons theory. We
prove that the cabling corresponds to taking tensor products of fundamental
representations of SU(2). This enables us to verify that the two apparently
distinct three-manifold invariants are equivalent for a specific relation of
the polynomial variables.
