# Geometric types of twisted knots

• [1] Department of Mathematics University of California at Santa Barbara Boston, MA 02215 USA
• [2] CMI, UMR 6632 du CNRS Université d’Aix-Marseille I 39, rue Joliot Curie F-13453 Marseille Cedex 13 FRANCE
• [3] Department of Mathematics Nihon University Tokyo 156-8550 JAPAN
• Volume: 13, Issue: 1, page 31-85
• ISSN: 1259-1734

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## Abstract

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Let $K$ be a knot in the $3$-sphere ${S}^{3}$, and $\Delta$ a disk in ${S}^{3}$ meeting $K$ transversely in the interior. For non-triviality we assume that $|\Delta \cap K|\ge 2$ over all isotopies of $K$ in ${S}^{3}-\partial \Delta$. Let ${K}_{\Delta ,n}$($\subset {S}^{3}$) be a knot obtained from $K$ by $n$ twistings along the disk $\Delta$. If the original knot is unknotted in ${S}^{3}$, we call ${K}_{\Delta ,n}$ a twisted knot. We describe for which pair $\left(K,\Delta \right)$ and an integer $n$, the twisted knot ${K}_{\Delta ,n}$ is a torus knot, a satellite knot or a hyperbolic knot.

## How to cite

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Aït-Nouh, Mohamed, Matignon, Daniel, and Motegi, Kimihiko. "Geometric types of twisted knots." Annales mathématiques Blaise Pascal 13.1 (2006): 31-85. <http://eudml.org/doc/10529>.

@article{Aït2006,
abstract = {Let $K$ be a knot in the $3$-sphere $S^3$, and $\Delta$ a disk in $S^3$ meeting $K$ transversely in the interior. For non-triviality we assume that $| \Delta \cap K | \ge 2$ over all isotopies of $K$ in $S^3 - \partial \Delta$. Let $K_\{\Delta , n\}$($\subset S^3$) be a knot obtained from $K$ by $n$ twistings along the disk $\Delta$. If the original knot is unknotted in $S^3$, we call $K_\{\Delta , n\}$ a twisted knot. We describe for which pair $(K, \Delta )$ and an integer $n$, the twisted knot $K_\{\Delta , n\}$ is a torus knot, a satellite knot or a hyperbolic knot.},
affiliation = {Department of Mathematics University of California at Santa Barbara Boston, MA 02215 USA; CMI, UMR 6632 du CNRS Université d’Aix-Marseille I 39, rue Joliot Curie F-13453 Marseille Cedex 13 FRANCE; Department of Mathematics Nihon University Tokyo 156-8550 JAPAN},
author = {Aït-Nouh, Mohamed, Matignon, Daniel, Motegi, Kimihiko},
journal = {Annales mathématiques Blaise Pascal},
keywords = {twisting pair; twisting number},
language = {eng},
month = {1},
number = {1},
pages = {31-85},
publisher = {Annales mathématiques Blaise Pascal},
title = {Geometric types of twisted knots},
url = {http://eudml.org/doc/10529},
volume = {13},
year = {2006},
}

TY - JOUR
AU - Aït-Nouh, Mohamed
AU - Matignon, Daniel
AU - Motegi, Kimihiko
TI - Geometric types of twisted knots
JO - Annales mathématiques Blaise Pascal
DA - 2006/1//
PB - Annales mathématiques Blaise Pascal
VL - 13
IS - 1
SP - 31
EP - 85
AB - Let $K$ be a knot in the $3$-sphere $S^3$, and $\Delta$ a disk in $S^3$ meeting $K$ transversely in the interior. For non-triviality we assume that $| \Delta \cap K | \ge 2$ over all isotopies of $K$ in $S^3 - \partial \Delta$. Let $K_{\Delta , n}$($\subset S^3$) be a knot obtained from $K$ by $n$ twistings along the disk $\Delta$. If the original knot is unknotted in $S^3$, we call $K_{\Delta , n}$ a twisted knot. We describe for which pair $(K, \Delta )$ and an integer $n$, the twisted knot $K_{\Delta , n}$ is a torus knot, a satellite knot or a hyperbolic knot.
LA - eng
KW - twisting pair; twisting number
UR - http://eudml.org/doc/10529
ER -

## References

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1. M. Aït Nouh, D. Matignon, K. Motegi, Obtaining graph knots by twisting unknots, C. R. Acad. Sci. Paris, Ser. I 337 (2003), 321-326 Zbl1033.57003MR2016983
2. M. Aït Nouh, D. Matignon, K. Motegi, Obtaining graph knots by twisting unknots, Topology Appl. 146-147 (2005), 105-121 Zbl1086.57008MR2107139
3. M. Culler, J. Luecke C. McA. Gordon, P. B. Shalen, Dehn surgery on knots, Ann. Math 125 (1987), 237-300 Zbl0633.57006MR881270
4. L. Glass, A combinatorial analog of the Poincaré Index Theorem, J. Comb. Theory Ser. B15 (1973), 264-268 Zbl0264.05112MR327558
5. H. Goda, C. Hayashi, H-J. Song, Dehn surgeries on $2$-bridge links which yield reducible $3$-manifolds
6. C. Goodman-Strauss, On composite twisted unknots, Trans. Amer. Math. Soc. 349 (1997), 4429-4463 Zbl0883.57004MR1355072
7. C.McA. Gordon, Combinatorial methods in Dehn surgery, Lectures at Knots 96 (1997), 263-290, World Scientific Publishing Co Zbl0940.57022MR1474525
8. C.McA. Gordon, R. A. Litherland, Incompressible planar surfaces in $3$-manifolds, Topology Appl. 18 (1984), 121-144 Zbl0554.57010MR769286
9. C.McA. Gordon, J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415 Zbl0678.57005MR965210
10. C.McA. Gordon, J. Luecke, Dehn surgeries on knots creating essential tori, I, Comm. Anal. Geom. 4 (1995), 597-644 Zbl0865.57015MR1371211
11. C.McA. Gordon, J. Luecke, Toroidal and boundary-reducing Dehn fillings, Topology Appl. 93 (1999), 77-90 Zbl0926.57019MR1684214
12. C.McA. Gordon, J. Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004), 417-485 Zbl1062.57006MR2074884
13. C. Hayashi, K. Motegi, Only single twist on unknots can produce composite knots, Trans. Amer. Math. Soc. 349 (1997), 4465-4479 Zbl0883.57005MR1355073
14. W. Jaco, P. B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 220 (1979) Zbl0415.57005MR539411
15. K. Johannson, Homotopy equivalences of $3$-manifolds with boundaries, (1979), Lect.Notes in Math, Springer-Verlag Zbl0412.57007MR551744
16. M. Kouno, K. Motegi, T. Shibuya, Twisting and knot types, J. Math. Soc. Japan 44 (1992), 199-216 Zbl0739.57003MR1154840
17. Y. Mathieu, Unknotting, knotting by twists on disks and property (P) for knots in ${S}^{3}$, Knots 90 (ed. by Kawauchi), Proc. 1990 Osaka Conf. on Knot Theory and Related Topics, de Gruyter (1992), 93-102 Zbl0772.57012MR1177414
18. W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), 37-44 Zbl0525.57003MR721450
19. K. Miyazaki, K. Motegi, Seifert fibered manifolds and Dehn surgery III, Comm. Anal. Geom. 7 (1999), 551-582 Zbl0940.57025MR1698388
20. J. Morgan, H. Bass, The Smith conjecture, (1984), Academic Press Zbl0599.57001MR758459
21. K. Motegi, Knot types of satellite knots and twisted knots, Lectures at Knots 96 (1997), 579-603, World Scientific Publishing Co Zbl0914.57005MR1474519
22. K. Motegi, T. Shibuya, Are knots obtained from a plain pattern always prime ?, Kobe J. Math. 9 (1992), 39-42 Zbl0766.57004MR1189955
23. Y. Ohyama, Twisting and unknotting operations, Rev. Mat. Univ. Complut. Madrid 7 (1994), 289-305 Zbl0861.57015MR1297516
24. D. Rolfsen, Knots and links, (1976), Publish or Perish, Berkeley, Calif. Zbl0339.55004MR515288
25. M. Scharlemann, Unknotting-number-one knots are prime, Invent. Math. 82 (1985), 37-55 Zbl0576.57004MR808108
26. M. Scharlemann, Producing reducible $3$-manifolds by surgery on a knot, Topology 29 (1990), 481-500 Zbl0727.57015MR1071370
27. M. Teragaito, Composite knots trivialized by twisting, J. Knot Theory Ramifications 1 (1992), 1623-1629 Zbl0765.57010MR1194998
28. W. P. Thurston, The geometry and topology of $3$-manifolds, (1979), Lecture notes, Princeton University
29. Y-Q. Wu, Dehn surgery on arborescent links, Trans. Amer. Math. Soc. 351 (1999), 2275-2294 Zbl0919.57008MR1458339

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