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 $\mathcal{P}\left(a_1,\dots ,a_n\right)$, we show that the Mahler measure of the Jones polynomial converges if all $a_i\to \infty$, and approaches infinity for $a_i=$ constant if $n\to \infty$, 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.