Recurrent generalization of f-polynomials for virtual knots and links

Amrendra Gill; Maxim Ivanov; Madeti Prabhakar; Andrei Vesnin

doi:10.3390/sym14010015

Recurrent generalization of f-polynomials for virtual knots and links

Amrendra Gill, Maxim Ivanov, Madeti Prabhakar, Andrei Vesnin

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Article number	15
Journal	Symmetry
Volume	14
Issue number	1
DOIs	https://doi.org/10.3390/sym14010015
Publication status	Published - Jan 2022

Keywords

Difference writhe
Flat virtual knot invariant
Virtual knot invariant

OECD FOS+WOS

1.02 COMPUTER AND INFORMATION SCIENCES
1.01 MATHEMATICS

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10.3390/sym14010015

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title = "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{\textquoteright}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.",

keywords = "Difference writhe, Flat virtual knot invariant, Virtual knot invariant",

author = "Amrendra Gill and Maxim Ivanov and Madeti Prabhakar and Andrei Vesnin",

note = "Funding Information: Author Contributions: Investigation, A.G., M.I., M.P. and A.V.; Methodology, A.G., M.I., M.P. and A.V.; Writing—original draft, A.G., M.I., M.P. and A.V.; Writing—review and editing, A.G., M.I., M.P. and A.V. The authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript Funding: A.G. and M.P. were supported by DST (project number DST/INT/RUS/RSF/P-02), M.I. was supported by RFBR (grant number 19-01-00569), and A.V. was supported by RSF (grant number 20-61-46005). Publisher Copyright: {\textcopyright} 2021 by the authors. Licensee MDPI, Basel, Switzerland.",

year = "2022",

month = jan,

doi = "10.3390/sym14010015",

language = "English",

volume = "14",

journal = "Symmetry",

issn = "2073-8994",

publisher = "Multidisciplinary Digital Publishing Institute (MDPI)",

number = "1",

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T1 - Recurrent generalization of f-polynomials for virtual knots and links

AU - Gill, Amrendra

AU - Ivanov, Maxim

AU - Prabhakar, Madeti

AU - Vesnin, Andrei

N1 - Funding Information: Author Contributions: Investigation, A.G., M.I., M.P. and A.V.; Methodology, A.G., M.I., M.P. and A.V.; Writing—original draft, A.G., M.I., M.P. and A.V.; Writing—review and editing, A.G., M.I., M.P. and A.V. The authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript Funding: A.G. and M.P. were supported by DST (project number DST/INT/RUS/RSF/P-02), M.I. was supported by RFBR (grant number 19-01-00569), and A.V. was supported by RSF (grant number 20-61-46005). Publisher Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/1

Y1 - 2022/1

N2 - 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.

AB - 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.

KW - Difference writhe

KW - Flat virtual knot invariant

KW - Virtual knot invariant

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JO - Symmetry

JF - Symmetry

SN - 2073-8994

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M1 - 15

ER -

Recurrent generalization of f-polynomials for virtual knots and links

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