A contribution of a U(1)-reducible connection to quantum invariants of links II: Links in rational homology spheres

TL;DR

This paper extends the U(1)-reducible connection contribution to quantum invariants of links in rational homology spheres, showing it as a formal power series with rational function coefficients and deriving a surgery formula relating it to colored Jones polynomials.

Contribution

It generalizes the U(1)-reducible connection contribution to links in rational homology spheres and establishes a surgery formula linking it to colored Jones polynomials.

Findings

The contribution is a formal power series in q-1 with rational function coefficients.

Coefficients are rational numbers with bounded denominators, as shown by Ohtsuki's theorem.

A surgery formula relates the contribution to the colored Jones polynomial.

Abstract

We extend the definition of the U(1)-reducible connection contribution to the case of the Witten-Reshetikhin-Turaev invariant of a link in a rational homology sphere. We prove that, similarly ot the case of a link in S^3, this contribution is a formal power series in powers of q-1, whose coefficients are rational functions of q^{color}, their denominators being the powers of the Alexander-Conway polynomial. The coefficients of the polynomials in numerators are rational numbers, the bounds on their denominators are established with the help of the theorem proved by T. Ohtsuki in Appendix 2. Similarly to the previously considered case of S^3, the U(1)-reducible connection contribution determines the trivial connection contribution into the Witten-Reshetikhin-Turaev invariant of algebraically connected links. We derive a surgery formula for the U(1)-reducible connection contribution, which relates it to the similar contribution into the colored Jones polynomial of a surgery link in S^3.