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Hilden, Hugh Michael and Lozano Imízcoz, María Teresa and Montesinos Amilibia, José María
(1985)
*Universal knots.*
In
Knot Theory and Manifolds.
Lecture Notes in Mathematics
(1144).
Springe, Berlin, pp. 25-59.
ISBN 978-3-540-15680-2

Official URL: http://link.springer.com/chapter/10.1007/BFb0075011

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http://link.springer.com/ | Publisher |

## Abstract

This paper contains detailed proofs of the results in the announcement "Universal knots'' [the authors, Bull. Amer. Math. Soc. (N.S.) 8 (1983), 449–450;]. The authors exhibit a knot K that is universal, i.e. every closed, orientable 3-manifold M can be represented as a covering of S3 branched over K, thereby giving an affirmative answer to a question of Thurston. The idea is to start with a 3-fold covering M→S3 branched over a knot and to change it to a covering M→S3 branched over a certain link L4 of four (unknotted) components. This shows that L4 is universal. Then a covering S3→S3 that is branched over a certain link L2 of two components with L4 in the preimage of L2, and a covering S3→S3 that is branched over K with L2 in the preimage of K, are constructed. This shows that L2 and K are universal. The knot K is rather complicated. In a later paper [Topology 24 (1985), no. 4, 499–504;] the authors show that the "figure eight'' knot is universal.

Item Type: | Book Section |
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Additional Information: | Proceedings of a Conference held in Vancouver, Canada, June 2–4, 1983 |

Uncontrolled Keywords: | universal knot; universal links |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 22072 |

Deposited On: | 24 Jun 2013 17:29 |

Last Modified: | 12 Dec 2018 15:14 |

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