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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
MathSciNet review: 507341
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Abstract: In this note it is shown that the longitudes of a $ \mu $-component homology boundary link L are in the second commutator subgroup G'' of the link group G if and only if the $ \mu $th Alexander ideal $ {\mathcal{E}_\mu }(L)$ is principal, generalizing the result announced for $ \mu = 2$ 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.

References [Enhancements On Off] (What's this?)

  • [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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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

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