Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T02:03:28.674Z Has data issue: false hasContentIssue false
  • English
  • Français

Examples of 3-knots with no 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]

Extract

We work throughout in the smooth category. Homomorphisms of fundamental and homology groups are induced by inclusion. An n-knot, form n ≥ 1, is an embedded n-sphere K ⊂ Sn+2. A Seifert manifold for K is a compact, connected, orientable (n + 1)-manifold V ⊂ Sn+2 with boundary ∂V = K. By [9] Seifert manifolds always exist. As in [9] let Y denote Sn+2 split along V; Y is a compact manifold with ∂Y = V0 ∪ V1, where Vt ≈ V. We say that V is a minimal Seifert manifold for K if π1Vt → π1Y is a monomorphism for t = 0, 1. (Here and throughout basepoint considerations are suppressed.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 3 , November 1991 , pp. 417 - 420
Copyright
Copyright © Cambridge Philosophical Society 1991

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]Baumslag, G.. A remark on generalized free products. Proc. Amer. Math. Soc. 13 (1962), 53–54.Google Scholar
[2]Burde, G. and Zieschang, H.. Eine Kennzeichnung der Torusknoten. Math. Ann. 167 (1966), 169–176.Google Scholar
[3]Burde, G. and Zieschang, H.. Knots (W. de Gruyter, 1985).Google Scholar
[4]Farrell, F. T.. The obstruction to fibering a manifold over a circle. Indiana J. Math. 21 (1971), 315–346.CrossRefGoogle Scholar
[5]Gutiérrez, M. A.. An exact sequence calculation for the second homotopy of a knot. Proc. Amer. Math. Soc. 32 (1972), 571–577.Google Scholar
[6]Gutiérrez, M. A.. An exact sequence calculation for the second homotopy of a knot, II. Proc. Amer. Math. Soc. 40 (1973), 327–330.Google Scholar
[7]Hillman, J.. 2-Knots and their Groups. Austral. Math. Soc. Lecture Series no. 5 (Cambridge University Press, 1989).Google Scholar
[8]Kervaire, M. A.. On higher dimensional knots. In Differential and Combinatorial Topology (Princeton University Press, 1965), pp. 105–109.Google Scholar
[9]Levine, J.. Unknotting spheres in codimension two. Topology 4 (1965), 9–16.Google Scholar
[10]Neuwirth, L. P.. Knot Groups. Annals of Math. Stud. no. 56 (Princeton University Press, 1965).CrossRefGoogle Scholar
[11]Robinson, B. J. S.. A Course in the Theory of Groups (Springer-Verlag, 1982).Google Scholar
[12]Swan, R. C.. Groups of cohomological dimension one. J. Algebra 12 (1969), 585–601.Google Scholar
[13]Yoshikawa, K.. Knot groups whose bases are abelian. J. Pure Appl. Math. 40 (1986), 321–335.Google Scholar
[14]Zeeman, E. C.. Twisting spun knots. Trans. Amer. Math. Soc. 115 (1965), 471–495.Google Scholar