Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T04:45:51.358Z Has data issue: false hasContentIssue false
  • English
  • Français

Manifolds with multiple knot-surgery descriptions

Published online by Cambridge University Press:  24 October 2008

W. R. Brakes
W. R. Brakes
Affiliation:
The Open University, Milton Keynes
Get access Rights & Permissions [Opens in a new window]

Extract

Every closed connected orientable three-manifold can be constructed by surgery along an appropriately chosen link in the three-sphere, S3 ((10), (15)). Such a link-surgery description is never unique, but the equivalence between different descriptions has been explicitly identified ((8),(3)). However, the situation with regard to manifolds that can be obtained by surgery along a knot in S3 is less clear ((6), p. 47): some manifolds are known to have at least two knot-surgery descriptions ((12), (14), p. 270, (1), (11)), whilst certain manifolds (e.g. S3 and S2 × S1) are widely suspected to have just one (see problems 1·15, 1·16 of (9)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 87 , Issue 3 , May 1980 , pp. 443 - 448
Copyright
Copyright © Cambridge Philosophical Society 1980

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)Akbulut, S.On 2-dimensional homology classes of 4-manifolds. Math. Proc. Cambridge Philos. Soc. 82 (1977), 99–106.CrossRefGoogle Scholar
(2)Dehn, M.Über die Topologie des dreidimensionalen Raumes. Math. Ann. 69 (1910), 137–168.CrossRefGoogle Scholar
(3)Fenn, R. and Rourke, C. P.On Kirby's calculus of links, Topology, 18 (1979), 1–16.CrossRefGoogle Scholar
(4)Fox, R. H. A quick trip through knot theory. In Topology of 3-manifolds (Prentice-Hall, 1962), pp. 120–167.Google Scholar
(5)Gordon, C. McA.Knots, homology spheres, and contractible manifolds. Topology 14 (1975), 151–172.CrossRefGoogle Scholar
(6)Gordon, C. McA. Some aspects of classical knot theory, Knot Theory, Springer-Verlag Lecture Notes, no. 685 (1978), 1–60.Google Scholar
(7)Gross, J.An infinite class of irreducible homology 3-spheres. Proc. Amer. Math. Soc. 25 (1970), 173–176.Google Scholar
(8)Kirby, R. C.A calculus for framed links in S 3. Invent. Math. 45 (1978), 35–56.CrossRefGoogle Scholar
(9)Kirby, R. C. Problems in low-dimensional manifold theory. Proc. A.M.S. Summer Inst. in Topology, Stanford, 1976.Google Scholar
(10)Lickorish, W. B. R.A representation of orientable combinatorial 3-manifolds. Ann. of Math. 76 (1962), 531–540.CrossRefGoogle Scholar
(11)Lickorish, W. B. R. Shake-slice knots. Topology of low-dimensional manifolds, Sussex (1977). Springer-Verlag Lecture Notes, no. 722 (1979), 67–70.Google Scholar
(12)Lickorish, W. B. R.Surgery on knots. Proc. Amer. Math. Soc. 60 (1977), 296–298.CrossRefGoogle Scholar
(13)Rolfsen, D. A surgical view of Alexander's polynomial, Geometric Topology. Springer Lecture Notes, no. 438 (1975), 415–423.CrossRefGoogle Scholar
(14)Rolfsen, D.Knots and Links (Publish or Perish Inc., 1976).Google Scholar
(15)Wallace, A. D.Modifications and cobounding manifolds. Can. J. Math. 12 (1960), 503–528.CrossRefGoogle Scholar