Topology
Twisted unknots
[Nœuds twistés]

Présenté par : Étienne Ghys

Aı̈t Nouh, Mohamed1 ; Matignon, Daniel2 ; Motegi, Kimihiko3
1 Department of Mathematics, University of California at Santa Barbara, CA 93106, USA
2 Université de Provence, CMI, 39, rue Joliot Curie, 13453 Marseille cedex 13, France
3 Department of Mathematics, Nihon University, Tokyo 156-8550, Japan
Comptes Rendus. Mathématique, Tome 337 (2003) no. 5, pp. 321-326.

Soient K un nœud dans la 3-sphère S3, et D un disque dans S3 rencontrant K transversalement dans son intérieur. Pour des raisons de non-trivialité, on peut supposer que |D∩K|⩾2 pour toutes les isotopies de K dans S3−∂D. Soit KD,n le nœud de S3 obtenu en effectuant n twists sur K le long du disque D. Si le nœud original K n'est pas noué dans S3, on dit que KD,n est un nœud twisté. Nous décrivons les paires (K,D) et les entiers n, pour lesquels le nœud twisté KD,n est un nœud torique, satellite, ou hyperbolique.

Let K be a knot in the 3-sphere S3, and D a disk in S3 meeting K transversely in the interior. For non-triviality we assume that |D∩K|⩾2 over all isotopies of K in S3−∂D. Let KD,n(⊂S3) be the knot obtained from K by n twisting along the disk D. If the original knot is unknotted in S3, we call KD,n a twisted unknot. We describe for which pairs (K,D) and integers n, the twisted unknot KD,n is a torus knot, a satellite knot or a hyperbolic knot.

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Accepté le :
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DOI : 10.1016/S1631-073X(03)00326-1
Aı̈t Nouh, Mohamed 1 ; Matignon, Daniel 2 ; Motegi, Kimihiko 3

1 Department of Mathematics, University of California at Santa Barbara, CA 93106, USA
2 Université de Provence, CMI, 39, rue Joliot Curie, 13453 Marseille cedex 13, France
3 Department of Mathematics, Nihon University, Tokyo 156-8550, Japan 
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Aı̈t Nouh, Mohamed; Matignon, Daniel; Motegi, Kimihiko. Twisted unknots. Comptes Rendus. Mathématique, Tome 337 (2003) no. 5, pp. 321-326. doi : 10.1016/S1631-073X(03)00326-1. http://www.numdam.org/articles/10.1016/S1631-073X(03)00326-1/

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