Writhe polynomials and shell moves for virtual knots and links
type: Item
instance of: scholarly article
title: Writhe polynomials and shell moves for virtual knots and links
zbMATH Open document ID: 1432.57023
author: Takuji Nakamura
author: Shin Satoh
author: Yasutaka Nakanishi
published in: European Journal of Combinatorics
publication date: 2019-11-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1905.03489
review text: In this paper, Nakamura, Nakanishi, and Satoh derive two results for virtual knots. The first result is that oriented virtual knots have the same writhe polynomial if and only if they are related by a set of moves which they call \textit{shell moves}. The second result is to show that two-component virtual links are classified up to shell moves in multiple ways using invariants of virtual links. The writhe polynomial of a virtual knot $K$, $W_K (t)$, was introduced by \textit{Z. Cheng} and \textit{H. Gao} in [J. Knot Theory Ramifications 22, No. 12, Article ID 1341002, 33 p. (2013; Zbl 1284.57005)]. It is given by the formula \[ W_K (t) = \sum_{n\neq 0} J_n (K)t^n-\sum_{n\neq 0} J_n (K) \] where $J_n(K)$ is the $n$-writhe of $K$. For more discussion of the $n$-writhe, see [\textit{S. Satoh} and \textit{K. Taniguchi}, Fundam. Math. 225, 327--341 (2014; Zbl 1302.57033)]. The authors introduce the notion of \textit{shell moves} for virtual knots and links. These are defined in terms of changes to the Gauss diagrams of the virtual links. The moves are illustrated in the following figure: {}\vspace{0.5cm} \begin{tikzpicture}[line width=1pt, >=Stealth] \draw [->] (0,-1) -- (0,1); \draw (0, 0) -- (1,0); \draw [<-] (1,-1) -- (1,1); \draw (0mm,4mm) arc [start angle=90, end angle=-90, radius=4mm]; \end{tikzpicture} \(\quad \overset{\mathrm{S}1}{\boldsymbol\sim} \quad\) \begin{tikzpicture}[line width=1pt, >=Stealth] \draw [->] (0,-1) -- (0,1); \draw (0, 0) -- (1,0); \draw [<-] (1,-1) -- (1,1); \draw (10mm,4mm) arc [start angle=-90, end angle=90, radius=-4mm]; \end{tikzpicture} \({}\qquad \qquad{}\) \begin{tikzpicture}[line width=1pt, >=Stealth] \draw [->] (0.0,0.0) -- (1.4,0.0); \draw (0.4,0.0) -- (0.4,1.0); \draw (1.0,0.0) -- (1.0,1.0); \end{tikzpicture} \(\quad \overset{\mathrm{S}2}{\boldsymbol\sim} \quad\) \begin{tikzpicture}[line width=1pt, >=Stealth] \draw [->] (0.0,0.0) -- (2.0,0.0); \draw (0.7cm,0.0cm) arc [start angle=0, end angle=180, radius=0.25cm]; \draw (1.6cm,0.0c…
reviewed by: Blake Winter
Mathematics Subject Classification ID: 57K12
zbMATH DE Number: 7138266
zbMATH Keywords: virtual knot
zbMATH Keywords: knot theory
MaRDI profile type: Publication
OpenAlex ID: W2981729094
arXiv ID: 1905.03489
cites work: A polynomial invariant of virtual knots
cites work: A polynomial invariant of virtual links
cites work: A linking number definition of the affine index polynomial and applications
cites work: Finite-type invariants of classical and virtual knots
cites work: Q3322863
cites work: On Knots. (AM-115)
cites work: Virtual knot theory
cites work: AN AFFINE INDEX POLYNOMIAL INVARIANT OF VIRTUAL KNOTS
cites work: A note on coverings of virtual knots
cites work: Q3418545
cites work: Minimal generating sets of Reidemeister moves
cites work: The writhes of a virtual knot
Wikidata QID: Q127010050
DOI: 10.1016/J.EJC.2019.103033
description: scientific article