Recurrent Generalization of F-Polynomials for Virtual Knots and Links

TL;DR

This paper introduces a recurrent method to construct new polynomial invariants for virtual links, generalizing F-polynomials, and demonstrates their superiority through explicit examples.

Contribution

It presents a novel recurrent construction of virtual link invariants using weight functions and smoothing, extending F-polynomials with improved distinguishing power.

Findings

New polynomial invariants are stronger than F-polynomials.

Explicit examples demonstrate the effectiveness of the new invariants.

The method generalizes existing invariants to broader classes of virtual links.

Abstract

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman's affine index polynomial and use smoothing in classical crossing of a virtual knot diagram. In this paper we introduce weight functions for ordered orientable virtual and flat virtual link. A flat virtual link is an equivalence class of virtual links in respect to a local symmetry changing type of classical crossing in a diagram. By considering three types of smoothings in classical crossings of a virtual link diagram and suitable weight functions, we provide a recurrent construction for new invariants. We demonstrate by providing explicit examples, that newly defined polynomial invariants are stronger than F-polynomials.