On the Mahler measure and root distribution of the $Q$-polynomial of links

Abstract: We study the roots and the Mahler measure of the $Q$-polynomial of links. We first consider links obtained by adding twists to a pair of parallel strands. We prove that the Mahler measure of the transformed $Q$-polynomial converges as the number of twists increases. We also show that all but a uniformly bounded number of distinct roots of the $Q$-polynomial approach the real interval $[-2,2]$. This behavior is different from that of the roots of the Jones polynomial under twisting. Numerical experiments on prime knots lead us to a conjecture about the roots of the transformed $Q$-polynomial of alternating knots. Finally, we compare real and unit-circle roots of the Alexander, Jones, and $Q$-polynomials, and give an infinite family of $2$-bridge links whose $Q$-polynomials have only real nonzero roots.