|
Organizers |
The Alexander Polynomial of a Rational Link
by
Kerry M. Luse
Trinity Washington University
Coauthors: Mark E. Kidwell
We relate some terms on the boundary of the Newton polygon of the Alexander polynomial Δ(x, y) of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize Δ(x, y) so that no x-1 or y-1 terms appear, but x-1Δ(x, y) and y-1Δ(x, y) have negative exponents, and so that terms of even total degree are positive and terms with odd total degree are negative. If the rational link has a reduced alternating diagram with no self crossings, then Δ(-1, 0) = 1. If the standard form of the rational link has m monochromatic twist sites, and the jth monochromatic twist site has q̂j crossings, then Δ(-1, 0) = ∏j=1m(q̂j+1). Our proof employs Kauffman's clock moves and a lattice for the terms of Δ(x, y) in which the y-power cannot decrease.
Date received: April 6, 2017
Copyright © 2017 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbof-04.