A virtualized skein relation for a multivariable polynomial invariant

TL;DR

This paper introduces a new skein relation for multivariable polynomials of virtual links, enabling analysis of virtual link properties through crossing modifications, advancing virtual knot theory.

Contribution

It provides a skein relation for multivariable polynomials of virtual links, extending the Jones polynomial framework to virtual crossings with specific restrictions.

Findings

Derived a skein relation for multivariable virtual link polynomials.

Analyzed properties of virtual links with crossing replacements.

Enhanced understanding of virtual link invariants.

Abstract

The virtual skein relation for the Jones polynomial of the virtual link diagram was introduced by N. Kamada, S. Nakabo, and S. Satoh. H. A. Dye, L. H. Kauffman, and Y. Miyazawa introduced multivariable polynomial, an invariant of virtual links, which is a refinement of Jones polynomial. In this paper, we give a skein relation for the multivariable polynomials among positive, negative, and virtual crossings with some restrictions. We apply this relation to study some properties of virtual links obtained by replacing a real crossing by a virtual crossing.