Publication: On knots that are universal
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Publication Date
1985
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Elsevier
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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
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R. H. Fox: A quick trip through knot theory. Topology of 3-manifolds and Related Topics. Prentice-Hall:Englewood Cliffs (1962).
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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).