On the Mahler measure of Jones polynomials under twisting

Abstract

We show that the Mahler measures of the Jones polynomial and of the colored Jones polynomials converge under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume under Dehn surgery. For pretzel links P(a 1 , . . . , a n ), we show that the Mahler measure of the Jones polynomial converges if all a i → ∞, and approaches infinity for a i = constant if n → ∞, just as hyperbolic volume. We also show that after sufficiently many twists, the coefficient vector of the Jones polynomial and of any colored Jones polynomial decomposes into fixed blocks according to the number of strands twisted.

The Mahler measure of algebraic numbers: a survey

“…Hironaka [85] has shown that among a wide class of Alexander polynomials of pretzel links, this one has the smallest Mahler measure. Champanerkar and Kofman [47] …”

Some families of links with divergent Mahler measure

“…In [4] and [8], the (Euclidean) Mahler measure of the Jones polynomial was investigated under twisting, while in [9] the Mahler measure of the multivariable Alexander polynomial was the subject. There are also several attempts such as [5,11] to find the locus of zeros for infinite families of knots and links.…”

“…Champanerkar and Kofman in [4] showed that the Mahler measure of the Jones polynomial of a link converges under twisting, and subsequently Silver and Williams showed [8] that a family of links with bounded twist number has bounded Mahler measure. One example of a family of links with divergent Mahler measure was given in [4]. There it was shown that the Mahler measure of the Jones polynomial of pretzel links diverges as the number of strands increases.…”

Euclidean Mahler measure and twisted links

“…Remark 3.2 (1) Champanerkar and Kofman showed in [5] using Jones-Wenzl idempotents that for full twists, V`q .t/ can be expressed as a rational function of t and t q . In fact, their result holds more generally for q full twists on any number of strands (with arbitrary orientation).…”