A polynomial invariant of virtual links

TL;DR

This paper introduces new polynomial invariants for virtual knots, links, and long flat virtual knots, generalizing existing invariants and exploring their relationships.

Contribution

It presents novel polynomial invariants for various virtual link types using parity axioms, extending previous invariants like the odd writhe and linking number.

Findings

New polynomial invariant for virtual knots using Manturov's parity axioms

Relation established between the new invariant and the affine index polynomial

Polynomial invariant for 2-component virtual links generalizing the linking number

Abstract

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author. The relation between this new polynomial invariant and the affine index polynomial is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.