Hostname: page-component-54dcc4c588-rz4zl Total loading time: 0 Render date: 2025-10-03T16:14:26.160Z Has data issue: false hasContentIssue false
  • English
  • Français

On the self-intersections of an immersed sphere

Published online by Cambridge University Press:  17 April 2009

Albert Borbély
Albert Borbély
Affiliation:
Department of Mathematics and Computer Science, Kuwait University, Safat 13060, Kuwait, e-mail: borbely@ mcs.sci.kuniv.edu.kw
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A closed curve f: S1 → ℝ2 in general position gives rise to a word whose letters are the self-intersection points, each of them appearing exactly twice. Such a word is called a Gauss code. The problem of determining whether a given Gauss code is realisable or not was first proposed by Gauss and it has been settled a long time ago. The analogous question for immersions f: S2 → ℝ3 in general position is settled only in a special case when the immersion has no triple points. We give a necessary condition for a system of curves to be realisable by a general immersion f: S2 → ℝ3.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 75 , Issue 3 , June 2007 , pp. 453 - 458
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]de Fraysseix, H. and de Mendez, O., ‘On a characterization of Gauss codes’, Discrete Comput. Geom. 22 (1999), 287–295.CrossRefGoogle Scholar
[2]Gauss, C.F., Werke VIII (Taubner, Leipzig, 1900).Google Scholar
[3]Lovász, L. and Marx, M.L., ‘A forbidden subgraph characterization of Gauss codes’, Bull. Amer. Math. Soc. 82 (1976), 121–122.CrossRefGoogle Scholar
[4]Lippner, G., ‘On double points of immersions of spheres’, Manuscripta Math. 113 (2004), 239–250.CrossRefGoogle Scholar
[5]Rosenstiehl, P., ‘Solution algébrique du probléme de Gauss sur la permutation des points d'intersection d'une ou plusieurs coubes fermées du plan’, C.R. Acad. Sci. Paris, Sér. A 283 (1976), 551–553.Google Scholar
[6]Rosenstiehl, P., ‘A new proof of the Gauss interlace conjecture’, Adv. in Appl. Math. 23 (1999), 3–13.CrossRefGoogle Scholar