By J. A. Hillman
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Extra resources for Alexander Ideals of Links
Subdeterminants Q if k < q and by ! if k ~ q. Crowell and Fox Eisenbud (We use the terminology of [13 ; page 573 ] and the (k+l) st Fitting [19 ] . is well known, system S in R. of the matrix This ideal is called the k th Clearly Ek(M) ~ Ek+I(M) multiplicative ideal of M is the ideal That it depends only on M, not and is proven for instance by Crowell and Ek(Ms) = Ek(M) S for any 28 For each k ~ 0 let AkM be the k th exterior power of M El4 ; page 507 ] and let ~k M = Ann AkM. The notation ~kM is due to Auslander and Buchsbaum E 5 ] who showed that if R is local and ~kM is principal for all k then M is a direct sum of cyclic modules, and used this to give criteria for projectivity.
Moreover (3 ) of these relations are the same for all p-component links. III If ~ = 2 the associated graded module G(B) = G(B) is a cyclic module over G(A) = G(fi) = ~ [ X 1 ..... Xn] , and Ann G(B) is generated by the "initial form" of the image of the Alexander polynomial AI(A) in ft. Thus the Chen groups of a 2-component link are effectively determined by its Alexander polynomial. (Here the initial form of a power series of fi is the homogeneous 52 polynomial in X. ) i IV The completed Crowell exact sequence 0 ---+ G'/G" ---+ A(L) ---+ I ---+ 0 of a link is invariant under 1-equivalence (hence under isotopy and concordance).
39 The column ideal class of M = coker Q is the row ideal class of coker Qtr, the cokernel of the transpose of Q. ) Q ~ Rp § coker Qtr § O and similarly (2) 0 ~ (coker Qtr)* ~ R p Q ~ R q ~ M ~ O. If Q2 is another presentation matrix for M, then it may be related to Q by "Tietze moves" the presentations isomorphic.  and on examining the effect of such moves on tr it may be seen that coker Qtr and coker Q2 are stably If Q is a monomorphism, then coker Qtr = elM and ~(M) = R. M ~ ! then y(M) is invertible.
Alexander Ideals of Links by J. A. Hillman