By J. A. Hillman

ISBN-10: 0387111689

ISBN-13: 9780387111681

ISBN-10: 3540111689

ISBN-13: 9783540111689

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D. A(L) ~ p - I . Since projective free, a torsion module has a short projective resolution A-modules are if and only if it has a square presentation matrix. If ~ known. = I, then ~(L) must be 1 also and the result is well If ~ = 2, the module G'/G" has a square presentation matrix even if ~(L) = 2. modules This was proven by Bailey who characterized arising from 2-component links as the A2-modules square presentation matrix of a particular form [ 7 ] [34 ] and Chapter VII). such admitting a (See also Cooper It may also he seen as follows.

Cochran's result extended to embeddings of arbitrary finite graphs and was published in [ 3 0 ] . In general H2(X;A) is free if and only if the projective dimension of A(L) is at most 2. d. A(L) ~ 2 , Schanuel's lemma applied to the exact sequence (2) implies that H2(X;A) is projective, and Suslin has shown that every projective A-module is free [185]. The argument in the other direction is obvious. For a boundary link a Mayer-Vietoris argument shows that H2(X;A) is free of rank ~ - I, but the 3-component homology boundary link of Figure V.!

With this row invariant may be characterized as follows. Theorem 12 The row ideal class p(M) is the isomorphismclass rank ] torsion free module Proof of the (~rM)/t(~rM). Let U be a (q-r)Xq submatrix of maximal rank q-r of the presentation matrix Q. Define ~: (Rq) r § R by ~(V 1 ,.. V 1 ,... v~rut~ 1 '''" V r 38 where the vectors V I ' " ' V r q • q matrix. in R q are used as the first r columns of a The map # is clearly alternating (for if two of the arguments V. V r) = 0 if any of the arguments V.

### Alexander Ideals of Links by J. A. Hillman

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