The Polynomials of Prime Virtual Knots of Genus 1 and Complexity at Most 5

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The Polynomials of Prime Virtual Knots of Genus 1 and Complexity at Most 5. / Vesnin, A. Y.; Ivanov, M. E.

In: Siberian Mathematical Journal, Vol. 61, No. 6, 11.2020, p. 994-1001.

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

BibTeX

@article{1a05ac88a57940b79e4738f1b8385be9,

title = "The Polynomials of Prime Virtual Knots of Genus 1 and Complexity at Most 5",

abstract = "Akimova and Matveev classified the prime virtual knots of genus 1 which admit diagrams with at most 5 classicalcrossings in 2017. In 2018,Kaur, Prabhakar, and Vesnin introduced the families of the $ L $- and$ F $-polynomials of virtual knots generalizing the Kauffman affine index polynomial.We introduce the notion of a totally flat-trivial virtual knot. We provethat the $ L $- and $ F $-polynomials for these knots coincide with the affine indexpolynomial. Also, we establish that all Akimova–Matveev knots are totally flat-trivialand calculate their affine index polynomials.",

keywords = "515.162.8, affine index polynomial, knot in a thickened torus, virtual knot",

author = "Vesnin, {A. Y.} and Ivanov, {M. E.}",

note = "Funding Information: The authors were supported by the Laboratory of Topology and Dynamics of Novosibirsk State University (Grant 14.Y26.31.0025 of the Ministry of Education and Science of the Russian Federation). Publisher Copyright: {\textcopyright} 2020, Pleiades Publishing, Ltd. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.",

year = "2020",

month = nov,

doi = "10.1134/S003744662006004X",

language = "English",

volume = "61",

pages = "994--1001",

journal = "Siberian Mathematical Journal",

issn = "0037-4466",

publisher = "MAIK NAUKA/INTERPERIODICA/SPRINGER",

number = "6",

}

RIS

TY - JOUR

T1 - The Polynomials of Prime Virtual Knots of Genus 1 and Complexity at Most 5

AU - Vesnin, A. Y.

AU - Ivanov, M. E.

N1 - Funding Information: The authors were supported by the Laboratory of Topology and Dynamics of Novosibirsk State University (Grant 14.Y26.31.0025 of the Ministry of Education and Science of the Russian Federation). Publisher Copyright: © 2020, Pleiades Publishing, Ltd. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2020/11

Y1 - 2020/11

N2 - Akimova and Matveev classified the prime virtual knots of genus 1 which admit diagrams with at most 5 classicalcrossings in 2017. In 2018,Kaur, Prabhakar, and Vesnin introduced the families of the $ L $- and$ F $-polynomials of virtual knots generalizing the Kauffman affine index polynomial.We introduce the notion of a totally flat-trivial virtual knot. We provethat the $ L $- and $ F $-polynomials for these knots coincide with the affine indexpolynomial. Also, we establish that all Akimova–Matveev knots are totally flat-trivialand calculate their affine index polynomials.

AB - Akimova and Matveev classified the prime virtual knots of genus 1 which admit diagrams with at most 5 classicalcrossings in 2017. In 2018,Kaur, Prabhakar, and Vesnin introduced the families of the $ L $- and$ F $-polynomials of virtual knots generalizing the Kauffman affine index polynomial.We introduce the notion of a totally flat-trivial virtual knot. We provethat the $ L $- and $ F $-polynomials for these knots coincide with the affine indexpolynomial. Also, we establish that all Akimova–Matveev knots are totally flat-trivialand calculate their affine index polynomials.

KW - 515.162.8

KW - affine index polynomial

KW - knot in a thickened torus

KW - virtual knot

UR - http://www.scopus.com/inward/record.url?scp=85100134133&partnerID=8YFLogxK

U2 - 10.1134/S003744662006004X

DO - 10.1134/S003744662006004X

M3 - Article

AN - SCOPUS:85100134133

VL - 61

SP - 994

EP - 1001

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 6

ER -

ID: 27709467