Geometric types of twisted knots

Mohamed Aït-Nouh[1]; Daniel Matignon[2]; Kimihiko Motegi[3]

  • [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

Annales mathématiques Blaise Pascal (2006)

  • Volume: 13, Issue: 1, page 31-85
  • ISSN: 1259-1734

Abstract

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Let K be a knot in the 3 -sphere S 3 , and Δ a disk in S 3 meeting K transversely in the interior. For non-triviality we assume that | Δ ∩ K | ≥ 2 over all isotopies of K in S 3 - ∂ Δ . Let K Δ , n ( ⊂ S 3 ) be a knot obtained from K by n twistings along the disk Δ . If the original knot is unknotted in S 3 , we call K Δ , n a twisted knot. We describe for which pair ( K , Δ ) and an integer n , the twisted knot K Δ , 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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