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 the dth component. Then the Mahler measure of the Alexander polynomial of l(q) converges to the Mahler measure of the Alexander polynomial of l as q goes to infinity, provided that some other component of l has nonzero linking number with the dth. Otherwise, the Mahler measure of the Alexander polynomial of l(q) has a well-defined bu different limiting behavior. Examples are given of links for which the Mahler measure of the Alexander polynomial is small. Possible connection with hyperbolic volume are discussed.