By J. A. Hillman
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Additional resources for Alexander Ideals of Links
As indicated in Chapter IV, Massey has also used c o m u t a t i v e algebra to extend Murasugi's equivalence (1) ~=#(2) to arbitrary link groups, and he has shown that the Chen groups of a 59 2-component link may be effectively computed from the Alexander polynomial [12~). For any group H, let H = H/H". naturally isomorphic. Then H/H H", H/(H)q and ~ q q are A mapf:H-~K induces maps f : H - ~ K and f :H/H -4 K/K , q q q and (~)q is naturally equivalent to (fq). We shall identify these naturally isomorphic quotients of H and naturally equivalent maps.
2. d. 10. // Part (i) of this Theorem was first proven by Torres who used properties of the Wirtinger presentation . 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 ). 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).
Furthermore if G is the group of a ~-component link L, then ~(L) = ~ if and only if the longitudes of L 55 lie in G(~). We shall also answer a question raised by Cochran (for 2-component links), by showing that H2(X;A ) = H2(X';~ ) projects onto H~X;~) if and only if E _I(L) = O. The Counter-example An epimorphism of groups f:G + H induces an epimorphism A(f):A(G) § A(H). Therefore if G is finitely generated, so the elementary ideals E,(G) = E,(A(G)) C ~ [G/G'] are defined, the image of El(G) in ~[H/H'] is contained in El(H).
Alexander Ideals of Links by J. A. Hillman