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

Lev Rozansky|ArXiv.org|Jun 11, 1998

Geometric and Algebraic Topology14 references5 citations

TL;DR

This paper extends the U(1)-reducible connection contribution to the Witten-Reshetikhin-Turaev invariant of links in rational homology spheres, proving it is a formal power series in $ q-1 $ with rational function coefficients whose denominators are powers of the Alexander-Conway polynomial. The coefficients in the numerators are rational numbers with bounded denominators, and the contribution determines the trivial connection part for algebraically split links, with a surgery formula linking it to the colored Jones polynomial in $ S^3 $.

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.

Motivation & Objective

• Extend the U(1)-reducible connection contribution to quantum invariants of links in rational homology spheres, generalizing prior results from $ S^3 $.
• Establish that the contribution is a formal power series in $ q-1 $, with coefficients that are rational functions of $ q^{\text{color}} $.
• Prove that the denominators of these rational functions are powers of the Alexander-Conway polynomial.
• Show that the numerators are rational numbers with bounded denominators, using Ohtsuki's theorem.
• Derive a surgery formula relating the U(1)-reducible contribution in rational homology spheres to the colored Jones polynomial in $ S^3 $.
• Establish that this contribution captures the trivial connection part for algebraically split links.

Proposed method

• The paper defines the U(1)-reducible connection contribution to the Witten-Reshetikhin-Turaev invariant in the context of rational homology spheres.
• It employs a formal power series expansion in $ q-1 $, with coefficients expressed as rational functions of $ q^{\text{color}} $.
• Using Ohtsuki's theorem from Appendix 2, it bounds the denominators of the rational coefficients in the numerators of these functions.
• A surgery formula is derived that relates the U(1)-reducible contribution in a rational homology sphere to the colored Jones polynomial of a surgery link in $ S^3 $.
• The contribution is shown to coincide with the trivial connection contribution for algebraically split links via topological and algebraic constraints.
• Analytic and topological techniques from quantum topology and 3-manifold theory are used to generalize the $ S^3 $-based results to rational homology spheres.

Experimental results

Research questions

• RQ1 How can the U(1)-reducible connection contribution to the Witten-Reshetikhin-Turaev invariant be generalized from $ S^3 $ to rational homology spheres?
• RQ2 What is the analytic structure of this contribution in terms of $ q-1 $, and what are the properties of its coefficients?
• RQ3 How do the denominators of the rational function coefficients relate to the Alexander-Conway polynomial in this setting?
• RQ4 What is the relationship between the U(1)-reducible contribution in a rational homology sphere and the colored Jones polynomial in $ S^3 $ via surgery?
• RQ5 Does the U(1)-reducible connection contribution capture the trivial connection part for algebraically split links in rational homology spheres?

Key findings

• The U(1)-reducible connection contribution to the Witten-Reshetikhin-Turaev invariant in a rational homology sphere is a formal power series in $ q-1 $.
• Each coefficient in this series is a rational function of $ q^{\text{color}} $, with denominators that are powers of the Alexander-Conway polynomial.
• The numerators of these rational functions are rational numbers, and their denominators are bounded by a constant independent of the link, as established via Ohtsuki's theorem.
• The contribution coincides with the trivial connection contribution for algebraically split links in rational homology spheres.
• A surgery formula is derived that expresses the U(1)-reducible contribution in a rational homology sphere as a sum over contributions from the colored Jones polynomial of a surgery link in $ S^3 $.
• The entire contribution is fully determined by the topological data of the link and the surgery description of the rational homology sphere.