Bing doubling and the colored Jones polynomial

TL;DR

This paper presents a formula to compute the reduced colored Jones polynomial of Bing doubles of knots, enabling new calculations and revealing divisibility properties of related quantum invariants of certain 3-manifolds.

Contribution

It introduces a novel formula linking Bing doubles' colored Jones polynomials to their companions, facilitating computations and theoretical insights into quantum invariants.

Findings

Derived a formula for the reduced colored Jones polynomial of Bing doubles.

Established a divisibility property of the Witten-Reshetikhin-Turaev invariant.

Applied results to analyze the Witten-Reshetikhin-Turaev invariant and Ohtsuki series.

Abstract

Bing doubling is an operation which gives a satellite of a knot. It is also applied to a link by specifying a component of the link. We give a formula to compute the reduced colored Jones polynomial of a Bing double by using that of the companion. This formula enables us to compute a lot of examples of the reduced colored Jones polynomial of Bing doubles. Moreover, from this formula we can derive a divisibility property of the unified Witten-Reshetikhin-Turaev invariant of integral homology spheres obtained by \pm1 surgery along Bing doubles of knots. This result is applied to the Witten-Reshetikhin-Turaev invariant and the Ohtsuki series of these integral homology spheres.