Milnor's invariants and the completions of link modules
Author:
Lorenzo Traldi
Journal:
Trans. Amer. Math. Soc. 284 (1984), 401-424
MSC:
Primary 57M05; Secondary 57M25
DOI:
https://doi.org/10.1090/S0002-9947-1984-0742432-3
MathSciNet review:
742432
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be a tame link of
components in
,
the abelianization of its group
, and
the augmentation ideal of the integral group ring
. The
-adic completions of the Alexander module and Alexander invariant of
are shown to possess presentation matrices whose entries are given in terms of certain integers
introduced by J. Milnor. Various applications to the theory of the elementary ideals of these modules are given, including a condition on the Alexander polynomial necessary for the linking numbers of the components of
with each other to all be zero. In the special case
, it is shown that the various Milnor invariants
are determined (up to sign) by the Alexander polynomial of
, and that this Alexander polynomial is 0 iff
for all
with
even; also, the Chen groups of
are determined (up to isomorphism) by those nonzero
with
minimal. In contrast, it is shown by example that for
the Alexander polynomials of a link and its sublinks do not determine its Chen groups.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1984-0742432-3
Keywords:
Tame links,
Milnor's invariants,
-adic completions,
Chen groups,
elementary ideals,
Alexander polynomials
Article copyright:
© Copyright 1984
American Mathematical Society