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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.
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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

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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