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Additional info for Alexander Ideals of Links
If At(L) # 0 then G'/G" has nontrivial p-torsion for p a prime integer if and only if p divides AI(L) , in which case p must divide the linking number AI(L )(1,1) (by the second Torres condition). 48 For Ann(G'/G") is generated by AI(L)/A2(L) , which is divisible by each of the prime factors of AI(L). Levine has shown that given any X in A 2 such that X = ~ there is a 2-component link L with linking number 0 such that AI(L) = X(t I - l)(t 2 - I) Ill7 ]. Hence on taking X to be a prime integer we see that G'/G" need not be torsion free as an abelian group.
Since these are true if Since be a link with group G ~ ~ I, n we may assume that large, B so and that ~ ~ 2. it will suffice to prove that is nilpotent, for G2 = G 3 So we may Let B=]]3, B = G'/G". e. that is finitely generated as an abel• B = B. ~ ~ 2. group, Massey's Theorem II implies that ^ has deficiency an abel• ~ 9 Since > 0 as a A-module, group if it is H2(G;~) = ~ (2) ~ 0. and so can only be finitely generated as Therefore G is a quotient of is abel• and so isomorphic to H2(X;~) = ~ - I , ~ ~ 2.
T3 - I ) ) and so for this link H2(X;A) is If L is unsplittable X is an Eilenberg-MacLane space K(G,I) for the group G (by the Sphere Theorem) and then Hq(G'; ~) = Hq(X;A). Thus in particular the commutator subgroup of a classical knot group has trivial integral homology in degree greater than I. The Crowell exact sequence for the free group F(~) is 0 ----+F(B)'/F(~)" ---+ A ~ ---+ A ---+ ~ ---+ 0 . The right hand terms constitute a partial resolution for the augmentation 44 module ~. We may obtain a complete equivariant homology of (SI) ~.