Twisted unknots.
type: Item
instance of: scholarly article
title: Twisted unknots.
zbMATH Open document ID: 1033.57003
DOI: 10.1016/S1631-073X(03)00326-1
author: Kimihiko Motegi
author: Mohamed Ait Nouh
author: Daniel Matignon
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-10-22T00:00:00Z
review text: Let \(D\) be a disk in the three-sphere and \(K\) a knot that meets \(D\) transversally in its interior. Let \(n\) be an integer and \(K_{D,n}\) the knot obtained from \(K\) by \(n\)-twisting along the disk \(D\). If \(K\) is trivial, then \((K,D)\) is called a twisting pair and the knot \(K_{D,n}\) is called a twisted unknot. It is well known from the Thurston uniformization theorem and the torus theorem that every knot in the three-sphere is a torus knot, a satellite knot or a hyperbolic knot. The paper under review discusses the geometric conditions under which a twisted unknot is a torus, satellite or hyperbolic knot. Namely, it is proved that: If \((K,D)\) is a hyperbolic pair, then \(K_{D,n}\) is a hyperbolic knot for any integer \(n\), such that \(| n| >1\). If \((K,D)\) is a Seifert fibered pair, then \(K_{D,n}\) is a torus knot for any integer \(n\). The paper considers also the case where \((K,D)\) is a toroidal pair, and gives a description of the pairs \((K,n)\) for which \(K_{D,n}\) is not a satellite knot.
reviewed by: Nafaa Chbili
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 1995409
zbMATH Keywords: twisted unknot
zbMATH Keywords: torus knot
zbMATH Keywords: satellite knot
zbMATH Keywords: hyperbolic knot
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1995409