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) -
(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,
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) .
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