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

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.