Links of oriented divides and fibrations in link exteriors
type: Item
instance of: scholarly article
title: Links of oriented divides and fibrations in link exteriors
zbMATH Open document ID: 1030.57007
author: Yanyan Li
published in: Osaka Journal of Mathematics
publication date: 2004-02-16T00:00:00Z
review text: An oriented divide is defined as the image of a self transverse immersion of a finite union of oriented circles into the unit disk \(D\). Considering the tangent bundle \(T D\) as a subset of \(T{\mathbb R}^2={\mathbb R}^4\) one gets \(TD=D\times{\mathbb R}^2\) and a divide introduces a map from the positive tangent half lines of the oriented circles into \(TD=D\times{\mathbb R}^2\). Intersecting the image of that map with the unit sphere \(S^3\subset{\mathbb R}^4\) one associates a link \(L(C)\subset S^3\) to each oriented divide. The authors prove that if \(D_L\subset D\) is a regular link projection of a link \(L\subset S^3\) then there exists an oriented divide \(C\), obtained from \(D_L\) by attaching small loops to it, such that \(L(C)\) is isotopic to \(L\), and a bound on the number of loops is given. (This theorem was proved in the setting of Legendrian knots by \textit{S. Chmutov, V. Goryunov}, and \textit{H. Murakami} [Math. Ann. 317, 389-413 (2000; Zbl 0957.57005)]). The authors also apply oriented divides to construct fibered knots with certain properties in the complement of a given link in \(S^3\).
reviewed by: Tobias Ekholm
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 1836042
zbMATH Keywords: link
zbMATH Keywords: oriented divide
zbMATH Keywords: fibered knots
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1836042