Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T07:28:28.007Z Has data issue: false hasContentIssue false
  • English
  • Français

On the existence of minimal Seifert manifolds

Published online by Cambridge University Press:  24 October 2008

Daniel S. Silver
Daniel S. Silver
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

For n ≥ 3, an n-knot K has a minimal Seifert manifold if and only if its group is isomorphic to an HNN-extension with finitely presented base. In this case, any Seifert manifold for K can be converted to a minimal Seifert manifold for K by some finite sequence of ambient 0- and 1-surgeries.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 103 - 109
Copyright
Copyright © Cambridge Philosophical Society 1993

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]Bieri, R. and Strebel, R.. Almost finitely presented soluble groups. Comment. Math. Helv. 53 (1978), 258–278.CrossRefGoogle Scholar
[2]Browder, W.. Structures on M × ℝ. Proc. Cambridge Philos. Soc. 61 (1965), 337–345.CrossRefGoogle Scholar
[3]Farber, M.. Minimal Seifert manifolds and the knot finiteness theorem. Israel J. Math. 66 (1989), 179–215.CrossRefGoogle Scholar
[4]Hillman, J.. 2-knots and their Groups. Australian Math. Soc. Lecture Series no. 5 (Cambridge University Press, 1989).Google Scholar
[5]Kervaire, M. and Weber, C.. A survey of multidimensional knots. In Hausmann, J. C. (ed.), Knot Theory, Lecture Notes in Math. vol. 685 (Springer-Verlag, 1978).CrossRefGoogle Scholar
[6]Kurosh, A. G.. The Theory of Groups, vol. 2 (Chelsea Publishing Co., 1960).Google Scholar
[7]Levine, J.. Unknotting spheres in codimension two. Topology 4 (1965), 9–16.CrossRefGoogle Scholar
[8]Lyndon, P. C. and Schupp, P. E.. Combinatorial Group Theory (Springer-Verlag, 1977).Google Scholar
[9]Neumann, B. H.. Some remarks on infinite groups. J. London Math. Soc. 12 (1937), 120–127.CrossRefGoogle Scholar
[10]Neuwirth, L. P.. Knot Groups. Annals of Math. Studies no. 56 (Princeton University Press, 1965).Google Scholar
[11]Rotman, J.. An Introduction to the Theory of Groups (Wm. C. Brown Publishers, 1988)Google Scholar
[12]Silver, D. S.. Examples of 3-knots with no minimal Seifert manifolds. Math. Proc. Cambridge Philos. Soc. 110 (1991), 417–420.CrossRefGoogle Scholar
[13]Strebel, R.. Finitely presented soluble groups. In Group Theory: Essays for Philip Hall (Academic Press, 1984), pp. 257–314.Google Scholar
[14]Yoshtkawa, K.. On n-knot groups which have abelian bases. Preprint.Google Scholar
[15]Yoshikawa, K.. Knot groups whose bases are abelian. J. Pure Appl. Algebra 40 (1986), 321–335.CrossRefGoogle Scholar