Mahler Measure of Alexander Polynomials

Abstract

Let l be an oriented link of d components in a homology 3-sphere. For any nonnegative integer q, let l(q) be the link of d−1 components obtained from l by performing 1/q surgery on its dth component ld. The Mahler measure of the multivariable Alexander polynomial Δl(q) converges to the Mahler measure of Δl as q goes to infinity, provided that ld has nonzero linking number with some other component. If ld has zero linking number with each of the other components, then the Mahler measure of Δl(q) has a well defined but different limiting behavior. Examples are given of links l such that the Mahler measure of Δl is small. Possible connections with hyperbolic volume are discussed.