Computation of Lickorish's Three Manifold Invariants using Chern-Simons Theory

TL;DR

This paper presents a simplified method for computing Lickorish's three-manifold invariants via Chern-Simons theory, linking cable computations to tensor products of SU(2) representations, and demonstrating their equivalence.

Contribution

It introduces an easier approach to compute bracket polynomials for cables using SU(2) representation theory, simplifying Lickorish's invariant calculations.

Findings

Simplified computation of bracket polynomials for cables.

Verification of equivalence between two three-manifold invariants.

Connection between cable operations and tensor products in SU(2).

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 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.