UNRAVELING THE INTEGRAL KNOT CONCORDANCE GROUP

5

C (Q) of homology (n-2)-connected rational knots is isomorphic to

C (Q) for e = (-l)n . Note that there is no condition placed on the

coefficients of the minimal polynomial of the endomorphism t .

The computation of C (Q) given by Levine [L2], employs the decom-

position theorem for Q[X]-modules/ for the endomorphism t induces a

Q[X]-module structure on the underlying vector space M . This techni-

que is applicable to any field but definitely does not apply to the

integral case. In face, this failure leads to a huge family of obstruc-

tions, which we will label "coupling invariants". These obstructions,

which form the second level of our computations, are expressed by the

following exact sequence:

0 - © cf.(Z) -

Ce

(X) - © C^(Q/Z) - C€(Q/Z) © © Coker c - 0

where C (Z) is the subgroup of C (Z) generated by isometric struc-

tures for which the minimal polynomial of the endomorphism t is cp(x) .

The summations are taken over all irreducible monic integral polynomials

c p (X) . This exact sequence reduces the computation of C (Z) to that

of the group C (Z) .

The study of C (Z) leads naturally to the study of the integral

domain S = Z [X]/(cp(X) ) • Unfortunately this ring is rarely integrally

closed and therefore is not always a Dedekind domain. (For example,

2

the polynomials cp(x) = X - X + b for b an integer, occur (by Lemma

1.11) in which case S in integrally closed if and only if 1 - 4b is

square free.) We also will need to permit non-projective torsion free

S-modules in our computation. In fact, we will show in Corollary 5.8

that certain elements of C (Z) can have no representatives on which

the underlying module is S-projective. The explicit computation of

C (Z) is made in Section Five, where we prove that the groups are

torsion-free in the symmetric case e = +1 (except for a single case

in which the group is of order two) and in the skew-symmetric case, that

the torsion subgroup generally consists of two-torsion whose 2-rank is

equal to one less than the number of "S-ramified" maximal ideals in S

(this is the number of primes p such that cp(x) s (X - 1/2)S mod p) .