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On knots that are universal

Montesinos Amilibia, José María and Hilden, Hugh Michael and Lozano Imízcoz, María Teresa (1985) On knots that are universal. Topology. An International Journal of Mathematics, 24 (4). pp. 49-504. ISSN 0040-9383

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Abstract

The authors construct a cover S3→S3 branched over the "figure eight" knot with preimage the "roman link" and a cover S3→S3 branched over the roman link with preimage containing the Borromean rings L. Since L is universal (i.e. every closed, orientable 3-manifold can be represented as a covering of S3 branched over L) it follows that the "figure eight'' knot is universal, thereby answering a question of Thurston in the affirmative. More generally, it is shown that every rational knot or link which is not toroidal is universal

Item Type:Article
Subjects:Sciences > Mathematics > Topology
ID Code:17185
References:

R. H. Fox: A quick trip through knot theory. Topology of 3-manifolds and Related Topics. Prentice-Hall:Englewood Cliffs (1962).

C. MCA. Gordon and W. Heil: Simply connected branched coverings of S’. Proc. Am. Math. Sot. 35 (1972), 287-288.

A. Hatcher and W. Thurston: Incompressible surfaces in 2-bridge knot complements. fnuent. Math. (to appear).

H. M. Hilden, M . T. Lozano and J. M. Montesinos:The Whitehead link, the Borromean ringsand the knot 946 are universal, Collectanea Mathematica, XXXIV (1983), pp. 19–28.

H. Schubertk: Knoten mit zwei Brücken. Math. Z. 65 (1956), 133-170.

W. Thurstonu: Universal links. (preprint, 1982).

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