The Alexander polynomial for Virtual Twist Knots

TL;DR

This paper introduces a family of virtual twist knots, derives a recursive formula for their Alexander polynomial, and provides evidence linking the polynomial to the odd writhe of virtual knots.

Contribution

It presents a new class of virtual knots and a recursive method to compute their Alexander polynomial, advancing understanding of virtual knot invariants.

Findings

Recursive formula for Alexander polynomial of virtual twist knots

Evidence linking Alexander polynomial to odd writhe

Extension of classical knot invariants to virtual knots

Abstract

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial $\Delta_0$ (as defined by Silver and Williams) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from $\Delta_0$.