Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T02:14:37.820Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperbolic, fibred links and fibre-concordances

Published online by Cambridge University Press:  24 October 2008

Teruhiko Soma
Teruhiko Soma
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

Let M be a closed, connected, orientable 3-manifold. In Row [10], Jaco and Myers [3] and Myers [7], it was pointed out that the topological type of M is closely related to the knot theory in M. Therefore it is an interesting problem to find knots in M with nice properties. Alexander proved M contains a fibred link (see [9]). Myers proved, in [7], M contains a hyperbolic knot, and, in [8], every link in M is concordant to a hyperbolic link. In this paper we consider the fibred version of his results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 2 , September 1984 , pp. 283 - 294
Copyright
Copyright © Cambridge Philosophical Society 1984

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]Harer, J.. How to construct all fibered knots and links. Topology 21 (1982), 263–280.CrossRefGoogle Scholar
[2]Jaco, W.. Lectures on Three-manifold topology. Conference Board of the Mathematical Sciences, no. 43 (Amer. Math. Soc., 1980).CrossRefGoogle Scholar
[3]Jaco, W. and Myers, R.. An algebraic determination of closed, orientable 3-manifolds. Trans. Amer. Math. Soc. 253 (1979), 149–170.CrossRefGoogle Scholar
[4]Kirby, R. C. and Lickorish, W. B. R.. Prime knots and concordance. Math. Proc. Cambridge Philos. Soc. 86 (1979), 437–441.CrossRefGoogle Scholar
[5]Magunus, W., Karrass, A. and Solitar, D.. Combinatorial Group Theory. Pure and Applied Applied Mathematics, vol. 13 (John Wiley, 1966).Google Scholar
[6] Menasco, W.. Closed incompressible surfaces in alternating knot and link complements. Topology 23 (1984), 37–44.CrossRefGoogle Scholar
[7] Myers, R.. Simple knots in compact, orientable 3-manifolds. Trans. Amer. Math. Soc. 273 (1982), 75–91.CrossRefGoogle Scholar
[8]Myers, R.. Homology cobordisms, link concordances, and hyperbolic 3-manifolds. Trans. Amer. Math. Soc. 278 (1983), 271–288.CrossRefGoogle Scholar
[9]Rolfsen, D.. Knots and Links (Publish or Perish Inc., 1976).Google Scholar
[10]Row, W. H.. An algebraic characterization of connected sum factors of closed 3-manifolds. Trans. Amer. Math. Soc. 250 (1979), 347–356.Google Scholar
[11]Soma, T.. Simple links and tangles. Tokyo J. Math. 6 (1983), 65–73.CrossRefGoogle Scholar
[12]Thurston, W.. Hyperbolic Structures on 3-Manifolds. Proc. Symp. on the Smith Conjecture. (Academic Press, to appear).Google Scholar
[13] Yamamoto, M.. Any fibered knot is concordant to a prime fibered knot in S3. Preprint.Google Scholar