Computation of Lickorish's Three Manifold Invariant Using ChernSimons Theory
Abstract
It is well known that any threemanifold can be obtained by surgery on a framed link in S^{3}. Lickorish gave an elementary proof for the existence of the threemanifold invariant of Witten using a framed link description of the manifold and the formalisation of the bracket polynomial as the TemperleyLieb Algebra. Kaul determined a threemanifold invariant from link polynomials in SU(2) ChernSimons 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) ChernSimons 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 threemanifold invariants are equivalent for a specific relation of the polynomial variables.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2000
 DOI:
 10.1007/s002200050014
 arXiv:
 arXiv:hepth/9901061
 Bibcode:
 2000CMaPh.209...29R
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Geometric Topology
 EPrint:
 25 pages, 11 eps figures, harvmac file (big mode)