Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T13:35:36.371Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinitely many disk knots with the same exterior

Published online by Cambridge University Press:  24 October 2008

Steven P. Plotnick
Steven P. Plotnick
Affiliation:
Columbia University, New York
Get access Rights & Permissions [Opens in a new window]

Extract

This paper addresses the question of how well the exterior of a knot determines the knot. It is well known that for a knotted sphere pair (Sn+2, Sn), n ≥ 2, a given exterior corresponds to at most two distinct knots (1), (4), (10); examples of distinct knots with the same exterior are given in (2), (6). For a knotted ball pair (Bn+2, Bn), n ≥ 2, the situation may be more complicated. In fact, given an arbitrary positive integer N, Hitt-Sumners (7), (8) constructed N distinct ball pairs with the same exterior, n ≥ 4.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 1 , January 1983 , pp. 67 - 72
Copyright
Copyright © Cambridge Philosophical Society 1983

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)Browder, W.Diffeomorphisms of 1-connected manifolds. Trans. AMS. 128 (1967), 155–163.CrossRefGoogle Scholar
(2)Cappell, S. and Shaneson, J. L.There exist inequivalent knots with the same complement. Ann. of Math. 103 (1976), 349–353.CrossRefGoogle Scholar
(3)Casson, A. J. and Harer, J. L.Some homology lens spaces which bound rational homology balls. Pacific J. of Math. 96 (1981), 23–36.CrossRefGoogle Scholar
(4)Gluck, H.The embedding of two-spheres in the four-sphere. Trans. AMS 104 (1962), 308–333.CrossRefGoogle Scholar
(5)Gordon, C. McA.Some higher-dimensional knots with the same homotopy groups. Quart. J. Math. Oxford (2), 24 (1973), 411–422.CrossRefGoogle Scholar
(6)Gordon, C. McA.Knots in the 4-sphere. Comment. Math. Helvetici 39 (1977), 585–596.Google Scholar
(7)Hitt, L. R. and Sumners, D. W.Many different disk knots with the same exterior. Comment. Math. Helvitici 56 (1981), 142–147.CrossRefGoogle Scholar
(8)Hitt, L. R. and Sumners, D. W.There exist arbitrarily many different disk knots with the same exterior. Proc. of AMS 86 (1982), 148–150.CrossRefGoogle Scholar
(9)Knapp, A. W.Doubly generated Fuchsian groups. Mich. Math. J. 15 (1968), 289–304.CrossRefGoogle Scholar
(10)Lashof, R. K. and Shaneson, J. S.Classification of knots in codimension two. Bull. AMS 75 (1969), 171–175.CrossRefGoogle Scholar
(11)Macbeath, A. M.Geometrical realization of isomorphisms between plane groups. Bull. AMS 71 (1965), 629–630.CrossRefGoogle Scholar
(12)Neumann, W. D. and Raymond, F. Automorphisms of Seifert manifolds (To appear).Google Scholar
(13)Pao, P. S.Non-linear circle actions on the 4-sphere and twisting spun knots. Topology 17 (1978), 291–296.CrossRefGoogle Scholar
(14)Plotnick, S.Embedding homology 3-spheres in S5, Pacific J. of Math. 101 (1982), 147–151.CrossRefGoogle Scholar
(15)Plotnick, S. The homotopy type of four-dimensional knot complements (Preprint).Google Scholar