Alexander Ideals of Links by J. A. Hillman

By J. A. Hillman

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There are short exact sequences I + Q(H;q) + H/Hq+IH" § H/HqH" + I by definition of the Chen groups, and so by the five lemma and induction a map f:H § K induces isomorphisms on all Chen groups if and only if all the maps fq are isomorphisms. The arguments below will be in terms of the groups H/HqH" excepting for one appeal to the computation of the Chen groups of a free group by Chen and Murasugi. The qth truncated Alexander module of H is the ~[H/H'~-module A (H) = ~ / ~ 2 ~ + ~ q ; q in ~articular A2(H) = ~/~2 is isomorghic to H/H' (see Chapter IV).

D 2 = A n and Dq = 0 for q > 2. the pair cell complex with is an exact sequence An ---+ d An+l ---+ A(L) ---+ 0 . nullity (2) of L, ~(L), i8 the rank of A(L) as a A-module. It is i m e d i a t e complex obtained (X',p-l(*)) so A(L) ~) ~ that ~(L) = min{k]Ek(L) by tensoring over A with ~ ~HI(X,* ;~) the cellular # 0} chain complex is just the cellular = ~. and is Therefore ~I. of the pair chain complex ~(E The (L)) = ~ of (X,*), and so 0, 43 ~(L) ~ ~. The Crowell sequence implies that rank G'/G" = ~(L) -i and tG'/G" = tA(L), while the exact sequence (2) implies that H2(X;A) is torsion free of rank ~ ( L ) - I.

The dimension of the vector space M / ~ M Let q be over the field R/~ . Then 2g ~k(M/~M) = 0 if k ~ q and eq+1 (M/~M) = R/~ , so Ok(M) ~ only if k ~ q. if and By Nakayama's le~m~a [ 4; page 21 2, M has a presentation with q generators. Since M / ~ M has dimension q, all the entries of the presentation matrix are in ~ , and hence Ek(M ) ~ and only if k < q, that is, if and only if ~k+|(M) ~ ~k(M) = n{~ prime I Ek(M) C ~ } = N{~ These results are well known. prime I ~k+l(M) ~ if In other words } = ~k+|(M).

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