no. 9-10
Topology/Dynamical Systems
A billiard containing all links
[Un billard réalisant tout entrelacs]

Présenté par : the Editorial Board

Dehornoy, Pierre1
1 UMPA, Ens de Lyon, 46, allée dʼItalie, 69364 Lyon, France
Comptes Rendus. Mathématique, Tome 349 (2011) no. 9-10, pp. 575-578.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.04.003
Dehornoy, Pierre 1

1 UMPA, Ens de Lyon, 46, allée dʼItalie, 69364 Lyon, France 
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Dehornoy, Pierre. A billiard containing all links. Comptes Rendus. Mathématique, Tome 349 (2011) no. 9-10, pp. 575-578. doi : 10.1016/j.crma.2011.04.003. http://www.numdam.org/articles/10.1016/j.crma.2011.04.003/

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[3] Lamm, C.; Obermeyer, D. 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] OʼRourke, J. Tying knots with reflecting lightrays, 2010 http://mathoverflow.net/questions/38813/

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

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