### Impacto

Montesinos Amilibia, José María and Hilden, Hugh Michael and Lozano Imízcoz, María Teresa
(1987)
*Nonsimple universal knots.*
Mathematical Proceedings of The Cambridge Philosophical Society, 102
(1).
pp. 87-95.
ISSN 0305-0041

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Official URL: http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=2484848

## Abstract

A link or knot in S 3 is universal if it serves as common branching set for all closed, oriented 3-manifolds. A knot is simple if its exterior space is simple, i.e. any incompressible torus or annulus is parallel to the boundary. No iterated torus knot or link is universal, but we know of many knots and links that are universal. The natural problem is to describe the class of universal knots, and this was asked by one of the authors in his address to the `Symposium of Kleinian groups, 3-manifolds and Hyperbolic Geometry' held in Durham, U. K., during July 1984. In the problem session of the same symposium W. Thurston asked if a non-simple knot can be universal and more concretely, if a cable knot can be universal. The question had the interest of testing whether the universality property has anything to do with the hyperbolic structure of some knots. That this is not the case is shown in this paper, where we give infinitely many examples of double, composite and cable knots that are universal.

Item Type: | Article |
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Uncontrolled Keywords: | link; knot; universal; 3-manifold; hyperbolic structure; cable knots |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 17163 |

References: | Hilden, H. M., Lozano, M. T. and Montesinos, J. M.. Universal knots. Knot Theory and Manifolds, Lecture Notes in Mathematics 1144 (Springer Verlag, 1985), 25–59. Hilden, H. M., Lozano, M. T. and Montesinos, J. M.. The Whitehead link, the Borromean rings and the knots 946 are universal. Collect. Math. 34 (1983), 19–28. Hilden, H. M., Lozano, M. T. and Montesinos, J. M.. On knots that are universal. Topology 24 (1985), 499–504. Hibsch, U.. Über offene Abbildungen auf die 3-sphare. Math. Z. 140 (1974), 203–230. Montesinos, J. M.. Sobre la Conjetura de Poincaré y los recubridores ramificados sobre un nudo. Ph.D. Thesis, Madrid (1971). Montesinos, J. M.. Reductión de la Conjetura de Poincaré a otras conjeturas geométricas. Revista Mat. Hisp.-Amer. (4) 32 (1972), 33–51. Thurston, W.. The Geometry and Topology of 3-manifolds. Lecture notes (Princeton) 1977–1978. Thurston, W.. Universal Links. (Preprint 1982.) Burde, G. and Zieschang, H.. Knots. (Walter de Gruyter, 1985). Seifert, H.. Topologie dreidimensionaler gefaserter Räume. Ada Math. 60 (1933), 147–288. |

Deposited On: | 23 Nov 2012 11:49 |

Last Modified: | 07 Feb 2014 09:43 |

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