Knotoids
type: Item
instance of: scholarly article
title: Knotoids
zbMATH Open document ID: 1271.57030
published in: Osaka Journal of Mathematics
publication date: 2012-04-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.4133
full work available at URL: https://projecteuclid.org/euclid.ojm/1332337244
reviewed by: Joel Foisy
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 6029033
zbMATH Keywords: knot diagram
zbMATH Keywords: theta-curve
zbMATH Keywords: knot group
zbMATH Keywords: crossing number
zbMATH Keywords: bracket polynomial
zbMATH Keywords: Seifert surface
zbMATH Keywords: skein algebra
author: Vladimir Turaev
describes a project that uses: KnotInfo
MaRDI profile type: Publication
arXiv ID: 1002.4133
cites work: Q3602397
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cites work: Prime decompositions of a \(\theta_n\)-curve in \(S^3\)
cites work: A prime decomposition theorem for handcuff graphs in \(S^{3}\)
cites work: Q4020209
cites work: A simple proof of the Murasugi and Kauffman theorems on alternating links
cites work: Q3827238
cites work: Q4720411
review text: In this paper, the author introduces the concept of \textit{knotoids}, which are represented by diagrams in a surface with the underlying curve a segment, rather than a circle. Knotoids are defined as equivalence classes of knotoid diagrams modulo the usual Reidemeister moves applied away from the endpoints. The author shows that knotoids in \(S^2\) generalize knots in \(S^3\) and defines a semigroup on knotoids in \(S^2\) that contains the usual semigroup of knots in the center. He uses techniques from the study of theta-curves to establish properties of this semigroup.NEWLINENEWLINEEvery knot in \(S^3\) may be represented by a knotoid diagram and every knotoid diagram in \(S^2\) determines a unique knot in \(S^3\) (actually two knots, see the paper for more details). Thus, studying knotoids may lead to more information about knots. For example, a Seifert surface can be constructed from a knotoid diagram in \(S^2\), and the author is able to use this construction to determine an upper bound on the genus of the associated knot. This upper bound is sometimes stronger than the usual upper bound on genus obtained from constructing the Seifert surface of the knot diagram.NEWLINENEWLINEThe paper also discusses an extension of several knot invariants to knotoids, including the knot group, crossing number, and the bracket polynomial. The author also introduces a skein algebra of knotoids in \(S^2\). In the final section, the author describes a surjective map between knotoids in \({\mathbb R}^2\) and knotoids in \(S^2\). He ends by giving examples of knotoids that are non-trivial in \({\mathbb R}^2\) but whose images are trivial in \(S^2\).
description: scientific article; zbMATH DE number 6029033