Comptes Rendus
Topology
Twisted unknots
[Nœuds twistés]
Comptes Rendus. Mathématique, Volume 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 |DK|⩾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 |DK|⩾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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DOI : 10.1016/S1631-073X(03)00326-1

Mohamed Aı̈t Nouh 1 ; Daniel Matignon 2 ; Kimihiko Motegi 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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Mohamed Aı̈t Nouh; Daniel Matignon; Kimihiko Motegi. Twisted unknots. Comptes Rendus. Mathématique, Volume 337 (2003) no. 5, pp. 321-326. doi : 10.1016/S1631-073X(03)00326-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00326-1/

[1] M. Aı̈t Nouh, D. Matignon, K. Motegi, Obtaining graph knots by twisting unknots, Topology Appl., in press

[2] M. Aı̈t Nouh, D. Matignon, K. Motegi, Satellite twisted unknots, in preparation

[3] H. Goda, C. Hayashi, H.-J. Song, private correspondence, 2002

[4] C. Goodman-Strauss On composite twisted unknots, Trans. Amer. Math. Soc., Volume 349 (1997), pp. 4429-4463

[5] C.McA. Gordon; J. Luecke Dehn surgeries on knots creating essential tori, I, Comm. Anal. Geom., Volume 4 (1995), pp. 597-644

[6] C.McA. Gordon; J. Luecke Toroidal and boundary-reducing Dehn fillings, Topology Appl., Volume 93 (1999), pp. 77-90

[7] C.McA. Gordon; J. Luecke Dehn surgeries on knots creating essential tori, II, Comm. Anal. Geom., Volume 8 (2000), pp. 671-725

[8] C.McA. Gordon, J. Luecke, Non-integral toroidal Dehn surgeries, Preprint

[9] W. Jaco; P.B. Shalen Seifert fibered spaces in 3-manifolds, Mem. Amer. Math. Soc., Volume 220 (1979)

[10] K. Johannson Homotopy Equivalences of 3-Manifolds with Boundaries, Lecture Notes in Math., 761, Springer-Verlag, 1979

[11] M. Kouno; K. Motegi; T. Shibuya Twisting and knot types, J. Math. Soc. Japan, Volume 44 (1992), pp. 199-216

[12] Y. Mathieu Unknotting, knotting by twists on disks and property (P) for knots in S3 (Kawauchi, ed.), Knots 90, Proc. 1990 Osaka Conf. on Knot Theory and Related Topics, de Gruyter, 1992, pp. 93-102

[13] W. Menasco Closed incompressible surfaces in alternating knot and link complements, Topology, Volume 23 (1984), pp. 37-44

[14] K. Miyazaki; K. Motegi Seifert fibered manifolds and Dehn surgery III, Comm. Anal. Geom., Volume 7 (1999), pp. 551-582

[15] The Smith Conjecture (J. Morgan; H. Bass, eds.), Academic Press, 1984

[16] K. Motegi; T. Shibuya Are knots obtained from a plain pattern always prime?, Kobe J. Math., Volume 9 (1992), pp. 39-42

[17] K. Motegi Knot types of satellite knots and twisted knots, Lectures at Knots 96, World Scientific, 1997, pp. 579-603

[18] Y. Ohyama Twisting and unknotting operations, Rev. Mat. Univ. Complut. Madrid, Volume 7 (1994), pp. 289-305

[19] D. Rolfsen Knots and Links, Publish or Perish, Berkeley, CA, 1976

[20] M. Scharlemann Producing reducible 3-manifolds by surgery on a knot, Topology, Volume 29 (1990), pp. 481-500

[21] M. Teragaito Composite knots trivialized by twisting, J. Knot Theory Ramifications, Volume 1 (1992), pp. 1623-1629

[22] W.P. Thurston The Geometry and Topology of 3-Manifolds, Lecture Notes, Princeton University, 1979

[23] Y.-Q. Wu Dehn surgery on arborescent links, Trans. Amer. Math. Soc., Volume 351 (1999), pp. 2275-2294

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