Lissajous curves as A'Campo divides, torus knots and their fiber surfaces
type: Item
instance of: scholarly article
title: Lissajous curves as A'Campo divides, torus knots and their fiber surfaces
zbMATH Open document ID: 1029.57006
author: Hiroshi Goda
author: Mikami Hirasawa
author: Yuichi Yamada
published in: Tokyo Journal of Mathematics
publication date: 2004-02-09T00:00:00Z
review text: N. A'Campo associated a link in \(S^3\) to a divide, a relative, generic immersion of a 1-manifold in a unit disk in \(\mathbb{R}^2\), and M. Hirasawa gave an algorithm to draw the link diagram of a divide. In this paper the authors give a visual proof that the link associated to the Lissajous curve (\(\cos p\theta\), \(\cos q\theta\)), regarded as a divide, is given by the torus link \(T(p,q)\). The approach to show this is to construct the link diagram and the fiber surface \(F\) for a Lissajous divide explicitly, where \(F\) is obtained as a smoothing of a long thin band having as many clasp-singularities as the unknoting number of \(T(p,q)\). The authors claim that such a construction of fiber surfaces is also available for any link arising from isolated complex curve singularities.
reviewed by: Masanori Morishita
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 1930978
zbMATH Keywords: A'Campo divides
zbMATH Keywords: torus knots
zbMATH Keywords: Lissajous curve
zbMATH Keywords: fiber surface
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208867
OpenAlex ID: W1980806005
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DOI: 10.3836/TJM/1244208867
description: scientific article; zbMATH DE number 1930978