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The nullity of a wild knot in a compact 3-manifold

Published online by Cambridge University Press:  09 April 2009

James M. McPherson
James M. McPherson
Affiliation:
School of General Studies Australian National UniversityCanberra.
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The nullity of the Alexander module of the fundamental group of the complement of a knot in S3 was one of the invariants of wild knot type defined and investigated by E. J. Brody in [1], in which he developed a generalised elementary divisor theory applicable to infinitely generated modules over a unique factorisation domain. Brody asked whether the nullity of a knot with one wild point was bounded above by its enclosure genus; for knots in S3, the present author showed in [6] that this was indeed the case. In [7], it was (prematurely) stated by the author that this was also the case for knots k embedded in a 3-manifold M so that H,(M — k) was torsion-free.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 3 , November 1973 , pp. 262 - 271
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Brody, E. J., ‘On infinitely generated modules’, Quart. J. Math. Oxford (2) 11 (1960), 141–150.CrossRefGoogle Scholar
[2]Brody, E. J., ‘The topological classification of the Lens spaces’, Ann. of Math. (2) 71 (1960), 163–184.CrossRefGoogle Scholar
[3]Crowell, R. H., ‘On the van Kampen theorem’, Pacific J. Math. 9 (1959), 43–50.CrossRefGoogle Scholar
[4]Fox, R. H., ‘Free differential calculus, II’, Ann. of Math. (2) 59 (1954), 196–210.CrossRefGoogle Scholar
[5]Fox, R. H., ‘Free differential calculus, V’, Ann. of Math. (2) 71 (1960), 408–422.CrossRefGoogle Scholar
[6]McPherson, J. M., ‘On the nullity and enclosure genus of wild knots’, Trans. Amer. Math. Soc. 144 (1969), 545–555.CrossRefGoogle Scholar
[7]McPherson, J. M., ‘Wild knots and arcs in a 3-manifold’, in Topology of Manifolds (ed. Cantrell, J. C. and Edwards, C. H. Jr), Markham Publishing Company, Chicago, 1970.Google Scholar
[8]Milnor, J., ‘A unique decomposition theorem for 3-manifolds’, Amer. J. Math. 84 (1962), 1–7.CrossRefGoogle Scholar
[9]Swarup, G. Anada, ‘Some properties of 3-manifolds with boundary’, Quart. J. Math. Oxford (2) 21 (1970), 1–23.CrossRefGoogle Scholar