An Orientation-Sensitive Vassiliev Invariant for Virtual Knots

TL;DR

This paper introduces a new first-order Vassiliev invariant derived from the Conway polynomial that can distinguish a virtual knot from its inverse, addressing an open question in knot theory.

Contribution

It provides the first example of an orientation-sensitive Vassiliev invariant for virtual knots, derived from the Conway polynomial.

Findings

The invariant distinguishes a virtual knot from its inverse.

The zeroth order invariant cannot distinguish virtual links from their inverses.

The zeroth order invariant vanishes for virtual knots.

Abstract

It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev invariant is derived from the Conway polynomial for virtual knots. Furthermore, it is shown that the zeroth order Vassiliev invariant coming from the Conway polynomial cannot distinguish a virtual link from its inverse and that it vanishes for virtual knots.