Comptes Rendus
Topology/Dynamical Systems
A billiard containing all links
Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 575-578.

We construct a 3-dimensional billiard realizing all links as collections of isotopy classes of periodic orbits. For every branched surface supporting a semi-flow, we construct a 3d-billiard whose collections of periodic orbits contain those of the branched surface. R. Ghrist constructed a knot-holder containing any link as collection of periodic orbits. Applying our construction to his example provides the desired billiard.

On construit un billard tridimensionnel réalisant tout entrelacs fini comme collection dʼorbites périodiques. Plus généralement, étant donné un patron, cʼest-à-dire une surface branchée munie dʼun semi-flot, on construit un billard dont la collection des orbites périodiques contient celle du patron. R. Ghrist a construit un tel patron contenant tous les entrelacs. On obtient le billard souhaité en appliquant notre construction à son exemple.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.04.003

Pierre Dehornoy 1

1 UMPA, Ens de Lyon, 46, allée dʼItalie, 69364 Lyon, France
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Pierre Dehornoy. A billiard containing all links. Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 575-578. doi : 10.1016/j.crma.2011.04.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.04.003/

[1] R.W. Ghrist Branched two-manifolds supporting all links, Topology, Volume 36 (1997) no. 2, pp. 423-488

[2] V.F.R. Jones; J.H. Przytycki Lissajous knots and billiard knots, Banach Center Publications, Volume 42 (1998), pp. 145-163

[3] C. Lamm; D. Obermeyer Billiard knots in a cylinder, J. Knot Theory Ramifications, Volume 8 (1999) no. 3, pp. 353-366

[4] H. Morton, in: Problems in Knots in Hellas ʼ98, vol. 2, Proceedings of the International Conference on Knot Theory and its Ramifications held in Delphi, August 7–15, 1998, pp. 547–559.

[5] J. OʼRourke Tying knots with reflecting lightrays, 2010 http://mathoverflow.net/questions/38813/

[6] S. Tabachnikov Billiards, Panoramas et Synthèses, vol. 1, Soc. Math. France, Paris, 1995 (vi+142 pp)

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