Alexander and writhe polynomials for virtual knots

TL;DR

The paper offers a new interpretation of the Alexander polynomial for virtual knots, demonstrating it determines the writhe polynomial and introducing a second-order writhe polynomial with applications.

Contribution

It provides a novel interpretation of the Alexander polynomial for virtual knots and introduces a second-order writhe polynomial with new applications.

Findings

Alexander polynomial determines the writhe polynomial for virtual knots

A second-order writhe polynomial is defined

New applications of writhe polynomials are presented

Abstract

We give a new interpretation of the Alexander polynomial $\Delta_0$ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, $\Delta_0$ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.