Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T19:48:02.663Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-simple universal knots

Published online by Cambridge University Press:  24 October 2008

H. M. Hilden ,
M. T. Lozano  and
J. M. Montesinos
H. M. Hilden
Affiliation:
University of Hawaii, U.S.A.
M. T. Lozano
Affiliation:
Universidad de Zaragoza, Spain
J. M. Montesinos
Affiliation:
Universidad Complutense de Madrid, Spain
Get access Rights & Permissions [Opens in a new window]

Extract

A link or knot in S3 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. Thurston gave the first examples of universal links [8], and subsequently we proved that all 2-bridge knots and links that can be universal (no torus knots or links) are in fact universal [3]. Some other universal knots are described in [1] and [2], together with a general procedure for constructing such knots. For a general reference to knots see [9].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 1 , July 1987 , pp. 87 - 95
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] 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.Google Scholar
[2] 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.Google Scholar
[3] Hilden, H. M., Lozano, M. T. and Montesinos, J. M.. On knots that are universal. Topology 24 (1985), 499–504.Google Scholar
[4] Hibsch, U.. Über offene Abbildungen auf die 3-sphare. Math. Z. 140 (1974), 203–230.Google Scholar
[5] Montesinos, J. M.. Sobre la Conjetura de Poincaré y los recubridores ramificados sobre un nudo. Ph.D. Thesis, Madrid (1971).Google Scholar
[6] Montesinos, J. M.. Reductión de la Conjetura de Poincaré a otras conjeturas geométricas. Revista Mat. Hisp.-Amer. (4) 32 (1972), 33–51.Google Scholar
[7] Thurston, W.. The Geometry and Topology of 3-manifolds. Lecture notes (Princeton) 1977–1978.Google Scholar
[8] Thurston, W.. Universal Links. (Preprint 1982.)Google Scholar
[9] Burde, G. and Zieschang, H.. Knots. (Walter de Gruyter, 1985).Google Scholar
[10] Seifert, H.. Topologie dreidimensionaler gefaserter Räume. Ada Math. 60 (1933), 147–288.Google Scholar