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Universal knots

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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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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
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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