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A non-homology boundary link with zero Alexander polynomial

Published online by Cambridge University Press:  17 April 2009

Jonathan A. Hillman
Affiliation:
Department of Pure Mathematics, Faculty of Arts, Australian National University, Canberra, ACT.
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Abstract

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This paper presents a necessary condition for a ribbon link to be an homology boundary link and gives a consequent simple counterexample to the conjecture of Smythe that the vanishing of the first Alexander polynomial characterizes homology boundary links among all 2-component links.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Atiyah, M.F., Macdonald, I.G., Introduction to commutative algebra (Addison-Wesley, Reading, Massachusetts; Menlo Park, California; London; Don Millas, Ontario; 1969).Google Scholar
[2]Bass, Hyman, “Libération des modules projectifs sur certains anneaux de polynômes”, Séminaire Bourbaki, 26e année, 1973/74, no. 448, 228254 (Lecture Notes in Mathematics, 431. Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
[3]Baumslag, Gilbert, “Groups with the same lower central sequence as a relatively free group. I: The groups”, Trans. Amer. Math. Soc. 129 (1967), 308321.Google Scholar
[4]Cochran, David S., “Links with zero Alexander polynomial” (PhD thesis, Dartmouth College, Hanover, New Hampshire, 1970).Google Scholar
[5]Crowell, R.H., “Corresponding group and module sequences”, Nagoya Math. J. 19 (1961), 2740.CrossRefGoogle Scholar
[6]Crowell, R.H., “Torsion in link modules”, J. Math. Mech. 14 (1965), 289298.Google Scholar
[7]Fox, R.H., “Some problems in knot theory”, Topology of 3-manifolds and related tcpics, 168176 (Proceedings of the University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1902).Google Scholar
[8]Hillman, Jonathan A., “Alexander ideals and Chen groups”, submitted.Google Scholar
[9]Kauffman, Louis H. and Taylor, Laurence R., “Signature of links”, Trans. Amer. Math. Soc. 216 (1976), 351365.CrossRefGoogle Scholar
[10]Murasugi, Kunio, “On a certain numerical invariant of link types”, Trans. Amer. Math. Soc. 117 (1965), 387422.CrossRefGoogle Scholar
[11]Neuwirth, L.P., Knot groups (Annals of Mathematics Studies, 56. Princeton University Press, Princeton, New Jersey, 1965).Google Scholar
[12]Quillen, Daniel, “Projective modules over polynomial rings”, Inventiones Math. 36 (1976), 167171.CrossRefGoogle Scholar
[13]Smythe, N., “Boundary links”, Topology seminar, Wisconsin, 1965, 6972 (Annals of Mathematics Studies, 60. Princeton University Press, Princeton, New Jersey, 1966).Google Scholar
[14]Sumners, D.W., “Invertible knot cobordisms”, Topology of manifolds, 200204 (Proceedings of the University of Georgia Topology of Manifolds Institute, 1969. Markham, Chicago, 1970).Google Scholar