The Multi-variable Affine Index Polynomial

TL;DR

This paper introduces a multi-variable extension of the Affine Index Polynomial for virtual links, generalizing previous invariants, and proves it is a Vassiliev invariant of order one, analyzing its behavior under component color shifts.

Contribution

It presents a new multi-variable Affine Index Polynomial for virtual links, extending existing invariants and establishing its Vassiliev order.

Findings

The invariant reduces to the original for virtual knots.

It generalizes Kauffman's recent virtual link invariant.

Proven to be a Vassiliev invariant of order one.

Abstract

We define a multi-variable version of the Affine Index Polynomial for virtual links. This invariant reduces to the original Affine Index Polynomial in the case of virtual knots, and also generalizes the version for compatible virtual links recently developed by L. Kauffman. We prove that this invariant is a Vassiliev invariant of order one, and study what happens as we shift the coloring of one or more components.