Homotopy and crossings of systems of curves on a surface
type: Item
instance of: scholarly article
title: Homotopy and crossings of systems of curves on a surface
zbMATH Open document ID: 0673.05027
DOI: 10.1016/0024-3795(89)90457-6
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://ir.cwi.nl/pub/10067
review text: Let \(C_ 1,...,C_ k\) and \(C_ 1',...,C_ k'\) be primitive closed curves on a compact orientable surface S. In the paper, a necessary and sufficient condition in terms of the number of crossings is found for the existence of a permutation \(\pi\) of \(\{\) 1,2,...,k\(\}\) such that \(C'_{\pi (i)}\) is freely homotopic to \(C_ i\) or \(C_ i^{-1}\), \(i\in \{1,2,...,k\}\). The condition is shown to be equivalent to the non- singularity of a certain infinite symmetric matrix. An important role in the proofs is played by a graph embedded on S made up of the curves \(C_ i'\).
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 57M15
zbMATH DE Number: 4101212
zbMATH Keywords: non-returning cycle
zbMATH Keywords: closed curves
zbMATH Keywords: graph
MaRDI profile type: Publication
OpenAlex ID: W1997645328
cites work: Decomposition of graphs on surfaces and a homotopic circulation theorem
author: Alexander Schrijver
description: scientific article; zbMATH DE number 4101212