Computation of Lickorish's Three Manifold Invariant Using Chern–Simons Theory

Abstract:

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 invariant of Witten using a framed link description of the manifold and the formalisation of the bracket polynomial as the Temperley–Lieb Algebra. Kaul determined a three-manifold invariant 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.