Longitudes of a link and principality of an Alexander ideal
Author:
Jonathan A. Hillman
Journal:
Proc. Amer. Math. Soc. 72 (1978), 370-374
MSC:
Primary 57M25
DOI:
https://doi.org/10.1090/S0002-9939-1978-0507341-6
MathSciNet review:
507341
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Abstract: In this note it is shown that the longitudes of a -component homology boundary link L are in the second commutator subgroup G'' of the link group G if and only if the
th Alexander ideal
is principal, generalizing the result announced for
by R. H. Crowell and E. H. Brown. These two properties were separately hypothesized as characterizations of boundary links by R. H. Fox and N. F. Smythe.
- [1] R. H. Crowell, Corresponding group and module sequences, Nagoya Math. J. 19 (1961), 27-40. MR 0140559 (25:3977)
- [2] -, Private communication to N. F. Smythe, May 1976.
- [3] M. A. Gutierrez, Polynomial invariants of boundary links, Rev. Colombiana Mat. VIII (1974), 97-109. MR 0367969 (51:4211)
- [4] H. W. Lambert, A 1-linked link whose longitudes lie in the second commutator subgroup, Trans. Amer. Math. Soc. 147 (1970), 261-269. MR 0267568 (42:2470)
- [5] D. Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, California, 1976. MR 0515288 (58:24236)
- [6] N. F. Smythe, Boundary links, Topology Seminar, Wisconsin, 1965, Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N. J., 1966, pp. 69-72.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1978-0507341-6
Keywords:
Homology boundary link,
longitude,
Seifert surface,
maximal abelian covering,
Mayer-Vietoris sequence,
second commutator subgroup,
Alexander ideal,
annihilator,
principal ideal
Article copyright:
© Copyright 1978
American Mathematical Society