Recurrent Generalization of F-Polynomials for Virtual Knots and Links

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.

[1]  Sofia Lambropoulou,et al.  Knotoids, Braidoids and Applications , 2017, Symmetry.

[2]  Louis H. Kauffman,et al.  Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory , 2014, 1409.2823.

[3]  Nicolas Petit.  The multi-variable Affine Index Polynomial , 2019, Topology and its Applications.

[4]  Julien Dorier,et al.  Topological Models for Open-Knotted Protein Chains Using the Concepts of Knotoids and Bonded Knotoids , 2017, Polymers.

[5]  Louis H. Kauffman,et al.  New invariants of knotoids , 2016, Eur. J. Comb..

[6]  Zhiyun Cheng.  The Chord Index, its Definitions, Applications, and Generalizations , 2016, Canadian Journal of Mathematics.

[7]  Keith Alexander,et al.  Proteins analysed as virtual knots , 2016, Scientific Reports.

[8]  Prabhakar Madeti,et al.  Arc shift number and region arc shift number for virtual knots , 2018, 1808.04172.

[9]  A. Vesnin,et al.  The Polynomials of Prime Virtual Knots of Genus 1 and Complexity at Most 5 , 2020, Siberian Mathematical Journal.

[10]  Yasutaka Nakanishi,et al.  The Intersection Polynomials of a Virtual Knot II: Connected Sums , 2023, Journal of Knot Theory and Its Ramifications.

[11]  L. Kauffman.  Virtual knot cobordism and the affine index polynomial , 2018, Journal of Knot Theory and Its Ramifications.

[12]  S. Panov,et al.  Representations of flat virtual braids which do not preserve the forbidden relations. , 2020, 2010.03162.

[13]  Susan G. Williams,et al.  WHAT IS...a Virtual Knot , 2017 .

[14]  Louis H Kauffman,et al.  The Affine Index Polynomial and the Sawollek Polynomial , 2020, Topology and Geometry.

[15]  S. Satoh,et al.  The writhes of a virtual knot , 2014 .

[16]  B. Mellor.  Alexander and writhe polynomials for virtual knots , 2016, 1601.07153.

[17]  Louis H. Kauffman.  Virtual Knot Theory , 1999, Eur. J. Comb..

[18]  K. Kaur,et al.  Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants , 2018, Journal of Knot Theory and Its Ramifications.

[19]  M. Ivanov,et al.  F-polynomials of tabulated virtual knots , 2019, 1906.01976.

[20]  A. Gill,et al.  Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves , 2019, 1908.05382.