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For all q ~ 2, there ~ chain complex to a finite free complex D o = A, D I = A n+! , 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) ~ ~.

2o Similarly, if G'/G" = 0 then ~ = ! ~ 0. d. d. ~ $ 2. d. 10. // Part (i) of this Theorem was first proven by Torres who used properties of the Wirtinger presentation [189]. If the commutator sub- group of a 2-component link is perfect, then AI(L) = I, so the linking number is • by the second Torres condition. the linking number is for G/G 3 of Chapter I. • (See Chapter Vll. That also follows from the Milnor presentation See also Chen [27]). In the knot theoretic 47 case (~= ]) the results of part (iii) were first obtained by Crowell [39 ] (Note that a Al-mOdule is pseudozero if and only if it is finite).

Hence G'/G" has a square presentation matrix. d. G'/G" ~ 2 since H2(X;A) ~(L) = 2) by Cochran's Theorem 3 nullity a. ---+ 0. Note also that is either 0 (if a(L) = I) or free (if lemma. I; an_d E (L) are both principal, (G' = G") then ~ ~ 2; or if G' is perfect 46 (iii) if e = ], then G'/G" has no nontrivial pseudozero submodule and Ann(G'/G") =(AI(L)/A2(L)); (iv) for each k ~ I, ek(tG'/G") ~ = (A~+k_] (L)/Ae+k(e)) 9 Proof (i) We may clearly assume that El(L) ~ 0, so that A(L) has rank I and A ( L ) / t A ( L ) ~ I.

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