On possible existence of HOMFLY polynomials for virtual knots

TL;DR

This paper explores the potential for defining HOMFLY polynomials for virtual knots, extending classical knot invariants to non-planar diagrams using a generalized Kauffman-Khovanov approach.

Contribution

It proposes a method to extend HOMFLY polynomials to virtual knots by generalizing the Kauffman-Khovanov calculus to non-planar diagrams, demonstrating topological invariance.

Findings

HOMFLY extension for virtual knots is possible

The generalized calculus preserves topological invariance

Initial examples support the existence of these polynomials

Abstract

Virtual knots are associated with knot diagrams, which are not obligatory planar. The recently suggested generalization from N=2 to arbitrary N of the Kauffman-Khovanov calculus of cycles in resolved diagrams can be straightforwardly applied to non-planar case. In simple examples we demonstrate that this construction preserves topological invariance -- thus implying the existence of HOMFLY extension of cabled Jones polynomials for virtual knots and links.