Recurrent Generalization of F-Polynomials for Virtual Knots and 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 the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram and suitable weight functions, there is provided a recurrent construction for new invariants. It is demonstrated by explicit examples that newly defined polynomial invariants are stronger than F-polynomials.

Local transformations and functorial maps

“…-- 2. Recurrent F -polynomials of virtual knots in [9] can be viewed as consecutive compositions of invariant derivations corresponding to the case T i = or and a polynomial invariant of flat knots.…”

“…A. Gill, M. Ivanov, M. Prabhakar and A. Vesnin used smoothings to construct multi-parameter series of virtual knot invariants [9].…”

Representations of Flat Virtual Braids by Automorphisms of Free Group

“…In the last decade many polynomial invariants of virtual knots and links have been introduced. Among them are affine index polynomial by Kauffman [33], writhe polynomial by Cheng and Cao [34], wriggle polynomial by Folwaczny and Kauffman [35], arrow polynomial by Dye and Kauffman [36], extended bracket polynomial by Kauffman [37], index polynomial by Im, Lee and Lee [38], zero polynomial by Jeong [39], sequences of L-polynomials and F-polynomials by Kaur, Prabhakar, and Vesnin [40] and recurrent generalizations of F-polynomials [41].…”