The Lehmer Polynomial and Pretzel Knots

E. Hironaka

The Lehmer polynomial, which is the monic, symmetric polynomial with smallest known Mahler measure, is also the Alexander polynomial for a (7,3,-2)-pretzel knot. In this paper, we find a formula for the Alexander polynomial for an infinite family of pretzel knots which includes the (7,3,-2)-pretzel knot. By generalizing this family, we obtain polynomials $\Delta_{p_1,\dots,p_k}(x)$, and show that in this larger family the Lehmer polynomial has smallest Mahler measure. The polynomials $\Delta_{p_1,\dots,p_k}(x)$ turn out to be the same as the denominators of growth functions of Coxeter reflection groups.