On isotopic weavings
type: Item
instance of: scholarly article
title: On isotopic weavings
zbMATH Open document ID: 1043.51013
author: A. A. Gaifullin
published in: Archiv der Mathematik
publication date: 2004-03-07T00:00:00Z
review text: A weaving \(W=({\mathcal L}, P, m)\) consists of a finite set \(\mathcal L\) of pairwise intersecting straight lines in real affine plane such that no three of these lines have a common intersection point, the set \(P\) of all intersection points of any two lines of \(\mathcal L\), and a map \(m : {\mathcal L} \times {\mathcal L} \setminus \{(k,k)\mid k \in {\mathcal L}\} \to \{-1,1\}\) with \(m(k,l) = -m(l,k)\). If \(m(l,k) = 1\), the line \(l\) is called to be locally above the line \(k\) at their intersection point \(l \wedge k \in P\). A weaving \(W\) is called realizable if there is a configuration \(\mathcal C\) of straight lines in real affine 3-space which can be projected orthogonally onto \({\mathcal L}\) via some projection \(\rho\) such that dist\(( \rho^{-1}(p) \cap l^*, p) > \text{ dist}( \rho^{-1}(p) \cap k^*, p)\) for any two lines \(l,k\) of \({\mathcal L}\) with \(p=l\wedge k\) and \(m(l,k) =1\), where \(k^*,l^* \in {\mathcal C}\) with \(\rho(k^*) = k\) and \(\rho(l^*) = l\). Two weavings \(W\) and \(W'\) are said to be isotopic if there is an isotopy of the plane moving the lines of \(W\) to the lines of \(W'\) such that all lines of \(W\) remain straight lines during this isotopy. The author proves that there exist isotopic weavings one of which is realizable and the other is not. This is achieved by giving a necessary and sufficient condition for a weaving isotopic to some special weaving defined by the author to be realizable.
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 51E30
zbMATH DE Number: 2051642
zbMATH Keywords: configurations of lines
zbMATH Keywords: isotopism
MaRDI profile type: Publication
full work available at URL: s00013-003-0099-3
OpenAlex ID: W2101918417
DOI: 10.1007/S00013-003-0099-3
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description: scientific article; zbMATH DE number 2051642