The Alexander polynomial of a rational link

Abstract

We relate some terms on the boundary of the Newton polygon of the Alexander polynomial [Formula: see text] of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize [Formula: see text] to be a true polynomial (as opposed to a Laurent polynomial), in such a way that terms of even total degree have positive coefficients and terms of odd total degree have negative coefficients. If the rational link has a reduced alternating diagram with no self-crossings, then [Formula: see text]. If the standard form of the rational link has [Formula: see text] monochromatic twist sites, and the [Formula: see text]th monochromatic twist site has [Formula: see text] crossings, then [Formula: see text]. Our proof employs Kauffman’s clock moves and a lattice for the terms of [Formula: see text] in which the [Formula: see text]-power cannot decrease.