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Some Counterexamples in Link Theory

Published online by Cambridge University Press:  20 November 2018

Dale Rolfsen
Dale Rolfsen*
Affiliation:
University of British Columbia, Vancouver, British Columbia
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This note is concerned specifically with links of two (disjoint) n-spheres in an (n + 2)-manifold M, i.e. embeddings L:Sn + Sn —> M. The links L0 and L1 are isotopic if they are t he ends of a continuous family Li:Sn + Sn —> M (0 ≦ t ≦ 1) of links. They are ambient isotopic or equivalent if there is a continuous family of self-homeomorphisms ht: M—> M (0 ≦ t ≦ 1) such that h0 = identity and h1◯L0 = L1. Ambient isotopic links are isotopic, but not conversely. For example, an isotopy can tie and untie little knots (as in Figure 3) in the components of a link, thus changing the original link into one which is inequivalent to the original.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 4 , August 1974 , pp. 978 - 984
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Alexander, J. W., An example of a simply connected surface bounding a region which is not simply connected, Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 8–10.Google Scholar
2. Bing, R. H., A simple closed curve that pierces no disk, J. Math. Pures Appl. 35 (1956), 337–43.Google Scholar
3. Hudson, J. F. P. and Sumners, D. W. L., Knotted ball pairs in unknotted sphere pairs, J. London Math. Soc. 14 (1966), 712–22.Google Scholar
4. Milnor, J., Isotopy of links, in algebraic geometry and topology, A symposium in honor of S. Lefschetz, Princeton, 1957, 280–306.Google Scholar
5. Rolfsen, D., Isotopy of links in codimension two, J. Indian Math. Soc. (to appear).Google Scholar
6. Zeeman, E. C., Linking spheres, Abh. Math. Sem. Univ. Hamburg 24 (1960), 149–52.Google Scholar