This article is cited in 1 paper

The polynomials of prime virtual knots of genus 1 and complexity at most 5

A. Yu. Vesnin^abc, M. E. Ivanov<sup>a

a</sup> Novosibirsk State University
^b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^c Tomsk State University

Abstract: Akimova and Matveev classified the prime virtual knots of genus 1 which admit diagrams with at most 5 classical crossings 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 prove that the $L$- and $F$-polynomials for these knots coincide with the affine index polynomial. Also, we establish that all Akimova–Matveev knots are totally flat-trivial and calculate their affine index polynomials.

Keywords: virtual knot, knot in a thickened torus, affine index polynomial.

UDC: 515.162.8

MSC: 35R30

Received: 06.08.2019
Revised: 06.08.2019
Accepted: 18.10.2019

DOI: 10.33048/smzh.2020.61.604

 English version: Siberian Mathematical Journal, 2020, 61:6, 994–1001

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