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Trivializing ribbon links by Kirby moves

Published online by Cambridge University Press:  17 April 2009

Jonathan A. Hillman
Jonathan A. Hillman
Affiliation:
Department of Pure Mathematics, School of General Studies, Australian National University, Canberra, ACT 2600, Australia.
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Abstract

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In this note it is shown that any ribbon link is a sublink of a ribbon link for which surgery on the longitudes gives a connected sum of copies of S1 x S2. In particular there are many links for which the analogue of the knot theoretic Property R fails, and sublinks of homology boundary links need not be homology boundary links. Higher dimensional analogues of these results are also given and it is shown that if n ≥ 2 the group of a ν-component ribbon n-link has a presentation of deficiency ν. Hence there are high dimensional slice knots which are not ribbon knots.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 21 , Issue 1 , February 1980 , pp. 21 - 28
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Baumslag, Gilbert, “Groups with the same lower central series as a relatively free group. II. Properties”, Trans. Amer. Math. Soc. 142 (1969), 507–538.CrossRefGoogle Scholar
[2]Fox, R.H., “A quick trip through knot theory”, Topology of 3-manifolds and related topics, 120–167 (Proc. University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1962).Google Scholar
[3]Fox, R.H., “Some problems in knot theory”, Topology of 3-manifolds and related topics, 168–176 (Proc. University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1962).Google Scholar
[4]Gutiérrez, M.A., “Boundary links and an unlinking theorem”, Trans. Amer. Math. Soc. 171 (1972), 491–499.CrossRefGoogle Scholar
[5]Gutiérrez, M.A., “Polynomial invariants of boundary links”, Rev. Colombiana Mat. 8 (1974), 97–109.Google Scholar
[6]Hillman, Jonathan A., “A non-homology boundary link with Alexander polynomial zero”, Bull. Austral. Math. Soc. 16 (1977), 229–236.CrossRefGoogle Scholar
[7]Hillman, Jonathan A., “Alexander ideals and Chen groups”, Bull. London Math. Soc. 10 (1978), 105–110.CrossRefGoogle Scholar
[8]Hitt, L.R., “Examples of higher-dimensional slice knots which are not ribbon knots”, preprint.Google Scholar
[9]Kervaire, Michael A., “Les nœuds de dimensions supérieures”, Bull. Soc. Math. France 93 (1965), 225–271.Google Scholar
[10]Kirby, Robion, “A calculus for framed links in S 3”, Invent. Math. 45 (1978), 35–56.CrossRefGoogle Scholar
[11]Kirby, Rob., “Problems in low dimensional manifold theory”, Algebraic and geometric topology, Part 2, 273–312 (Proc. Symposia in Pure Mathematics, 32. American Mathematical Society, Providence, Rhode Island, 1978).CrossRefGoogle Scholar
[12]Kirby, Robion and Melvin, Paul, “Slice knots and property R”, Invent. Math. 45 (1978), 57–59.CrossRefGoogle Scholar
[13]Poenaru, Valentin, “A note on the generators for the fundamental group of the complement of a submanifold of codimension 2”, Topology 10 (1971), 47–52.CrossRefGoogle Scholar
[14]Rubinstein, J.H., “Dehn's lemma and handle decomposition of some 4-manifolds” (Mathematics Research Report, 1–1979. University of Melbourne, Parkville, Victoria, 1979).Google Scholar
[15]Smythe, N., “Boundary links”, Topology Seminar, Wisconsin, 1965, 69–72 (Annals of Mathematics Studies, 60. Princeton University Press, Princeton, New Jersey, 1966).Google Scholar
[16]Smythe, N., “n-linking and n-splitting”, Amer. J. Math. 92 (1970), 272–282.CrossRefGoogle Scholar
[17]Stallings, John, “Homology and central series of groups”, J. Algebra 2 (1965), 170–181.CrossRefGoogle Scholar
[18]Yajima, Takeshi, “On a characterization of knot groups of some spheres in R 4”, Osaka J. Math. 6 (1969), 435–446.Google Scholar
[19]Yanagawa, Takaaki, “On ribbon 2-knots. The 3-manifold bounded by the 2-knots”, Osaka J. Math. 6 (1969), 447–464.Google Scholar