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.
Highlights
The theory of virtual knots and links was introduced by Kauffman in [1] as a generalization of classical knot theory
We use fspann,k ( L) to construct a family of oriented virtual knot invariants Fen,k,m (t, `, v) on variables t, ` and v in Theorem 6 and demonstrate that these 3-variable polynomials are stronger than the F-polynomial
We define odd and even weight functions associated with classical crossings in diagrams of virtual knots
Summary
The theory of virtual knots and links was introduced by Kauffman in [1] as a generalization of classical knot theory. Such invariants, corresponding to type-2 smoothing, are constructed in Corollary 1 and used in Example 5 to demonstrate that the virtual Kishino knot, a famous connected sum of two trivial virtual knots, is non-trivial. We use fspann,k ( L) to construct a family of oriented virtual knot invariants Fen,k,m (t, `, v) on variables t, ` and v in Theorem 6 and demonstrate that these 3-variable polynomials are stronger than the F-polynomial
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