By J. A. Hillman
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Additional info for Alexander Ideals of Links
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. ) the unknot or // CHAPTER V THE VANISHING OF ALEXANDER IDEALS At the 1961 Georgia conference on Topology of 3-Manifolds, Fox raised the question of the geometric significance of the identical vanishing of the first Alexander polynomial of a 2-component link [50; Problem 16~.
2 If B = 2 the module ZZ O - ~ + I B has a square presentation matrix with determinant generating E(Eo(B)) and so divisible by C(Ao(B)) = e(Al(A)) , and thus 2Z O B = 0 only if e(Al(A)) = _+ I. Crowell showed also that (if ~ = 2 and EXtA(B,I) ~ 7Z/e(AI(A)) and asked whether the class of the extension could be used to distinguish That Z g ~ A 1 (A) # 0) between two links. I . IP _C Ek_|(B) for ! < ~. I-P) . I, thus proving part of the Crowell-Strauss result. We shall sketch a proof of (a) and of part of (b).
Let M be a finitely generated R-module. is the dimension of the vector space M o = R o ~ R fractions of R. m = 0 for some nonzero r in R } , and M is an R-torsion module if tM = 0. annihilator ideal of M is A n n M = ~r in R [ r . m The = 0 for all m in M } . Let RP Q be a finite presentation Rq _ ~ for M. M ___+ 0 This presentation has deficiency q - p , and is said to give a short free resolution of M if the map Q is injective. For each k ~ 0 the k th elementary Ek(M) generated by the (q-k)• representing determinantal [43 ; page 101] .
Alexander Ideals of Links by J. A. Hillman