# Virtual Parity Alexander Polynomial

## TL;DR

This paper introduces the parity virtual Alexander polynomial, an invariant for virtual knots, exploring its properties and demonstrating its ability to distinguish knots that cannot be unknotted by odd crossing changes.

## Contribution

It defines a new invariant for virtual knots based on parity, extending previous work and providing tools to analyze virtual knot unknottability.

## Key findings

- The invariant is computed for various examples.
- Many virtual knots cannot be unknotted by changing only odd crossings.
- The properties of the invariant are systematically explored.

## Abstract

In this paper, we define the parity virtual Alexander polynomial following the work of BDGGHN [1] and Kaestner and Kauffman [10]. The properties of this invariant are explored and some examples are computed. In particular, the invariant demonstrates that many virtual knots can not be unknotted by crossing change on only odd crossings.