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The geometry of the slice-ribbon problem

Published online by Cambridge University Press:  24 October 2008

Joel Hass
Joel Hass
Affiliation:
University of California, Berkeley
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Extract

One of the principal problems in low-dimensional topology concerns the question of whether all slice knots are ribbon(3),(4). It will be shown in this paper that this problem is closely related to geometric properties of discs properly embedded in the unit 4-ball in Euclidean space. The main result, Theorem 1·13, states that a knot is ribbon if and only if a representative within its isotopy class bounds an embedded minimal disc, and that this in turn happens if and only if a representative bounds a disc of sufficiently small total curvature.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 1 , July 1983 , pp. 101 - 108
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

(1)Almgren, F. and Solomon, B. Personal Communication. See also Abstracts of the A.M.S., April 1980.Google Scholar
(2)Chern, S. S. and Lashof, R. K.On the total curvature of an immersed manifold I. Amer. J. Math. 79 (1957), 306–318.CrossRefGoogle Scholar
(3)Fox, R. H.A quick trip through knot theory. Proc. Univ. Georgia Inst. 1962, pp. 120–167.Google Scholar
(4)Kirby, R. Problems in low dimensional manifold theory. Proc. A.M.S. Summer Institute in Topology. Stanford, 1976.Google Scholar
(5)Osserman, R.A survey of minimal surfaces, Van Nostrand Rheinhold Mathematical Studies, no. 25 (1969).Google Scholar
(6)Langevin, R. and Rosenberg, H.On curvature integrals and knots. Topology 15 (1976), 405–416.CrossRefGoogle Scholar
(7)Milnor, J.On the total curvature of knots. Ann. Math. 52 (1950), 248–257.CrossRefGoogle Scholar
(8)Meeks, W. H. and Yau, S. T.Topology of embedded minimal surfaces and the problem of uniqueness. Math. Z. 179 (1982), 151–168.CrossRefGoogle Scholar