A Contribution of the Trivial Connection to the Jones Polynomial and Witten's Invariant of 3d Manifolds II

TL;DR

This paper extends previous work on the trivial connection contribution to the Jones polynomial and Witten's invariant from knots to links, establishing new formulas and relations, and applying them to rational homology spheres.

Contribution

It introduces a link formula for the Jones polynomial based on Reshetikhin's work, relating it to the Alexander polynomial, and derives a surgery formula for Witten's invariant.

Findings

Derived a formula for the Jones polynomial of links using Reshetikhin's approach.

Established a relation between Reshetikhin's parameters and the multivariable Alexander polynomial.

Provided a link surgery formula for Witten's invariant of rational homology spheres.

Abstract

We extend the results of our previous paper from knots to links by using a formula for the Jones polynomial of a link derived recently by N. Reshetikhin. We illustrate this formula by an example of a torus link. A relation between the parameters of Reshetikhin's formula and the multivariable Alexander polynomial is established. We check that our expression for the Alexander polynomial satisfies some of its basic properties. Finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to Witten's invariant of rational homology spheres.