no. 5
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.

Reçu le :
Accepté le :
Publié le :
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 
@article{CRMATH_2003__337_5_321_0,
     author = {A{\i}\ensuremath{\ddot{}}t Nouh, Mohamed and Matignon, Daniel and Motegi, Kimihiko},
     title = {Twisted unknots},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {321--326},
     publisher = {Elsevier},
     volume = {337},
     number = {5},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00326-1},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00326-1/}
}
TY  - JOUR
AU  - Aı̈t Nouh, Mohamed
AU  - Matignon, Daniel
AU  - Motegi, Kimihiko
TI  - Twisted unknots
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 321
EP  - 326
VL  - 337
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(03)00326-1/
DO  - 10.1016/S1631-073X(03)00326-1
LA  - en
ID  - CRMATH_2003__337_5_321_0
ER  - 
%0 Journal Article
%A Aı̈t Nouh, Mohamed
%A Matignon, Daniel
%A Motegi, Kimihiko
%T Twisted unknots
%J Comptes Rendus. Mathématique
%D 2003
%P 321-326
%V 337
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00326-1/
%R 10.1016/S1631-073X(03)00326-1
%G en
%F CRMATH_2003__337_5_321_0
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/

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

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

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

[7] Gordon, C.McA.; Luecke, J. 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] Jaco, W.; Shalen, P.B. Seifert fibered spaces in 3-manifolds, Mem. Amer. Math. Soc., Volume 220 (1979)

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :