The W-polynomial and the Mahler Measure of the Kauffman Bracket

TL;DR

This paper explores the relationship between the W-polynomial, the Mahler measure, and the Kauffman bracket, providing bounds, formulas, and divergence conditions for specific link families.

Contribution

It introduces a geometric bound on the Mahler measure of the Kauffman bracket and derives a general formula for links obtained by rational tangle surgeries.

Findings

A link diagram's geometric property bounds the Mahler measure.

A general form for the Kauffman bracket after rational tangle surgery is established.

Conditions for divergence of the Mahler measure in certain link families are identified.

Abstract

The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler measure of the Kauffman bracket. Second we find a general form for the Kauffman bracket of a link found by surgering in a single rational tangle along n unlinked components, all in a particular annulus. We then give a condition under which the Mahler measure of the Kauffman bracket of such families diverges. We give examples of the condition in action.