We also give examples of computations of the invariant SC (-1) . The
constructions (given preceding (4.15)) employs the Hilbert class field
and the invariant SC(-l) is related to the existence of a non-principal
prime ideal in F which becomes principal in a quadratic extension E
of F contained in the Hilbert class field of F .
The first part of Section Five, as previously noted, is devoted to
the application of the computation of Section Four to the orders arising
in the knot concordance group. We then proceed to study algebraically
three geometrically interesting subgroups of the knot concordance group:
the cyclotomic subgroup, the quadratic subgroup and the fibered knots.
The first subgroup is specified algebraically by requiring that the
endomorphism t be invertible and that the minimal polynomial of s =
1 - t (which is then an isometry of the inner product associated to
t . This is the reason for the name isometric structure.) be a product
of cyclotomic polynomials. By a theorem of Grothendieck (See Milnor
[M2]), this is the case for the knots arising from the isolated singu-
larities of complex hypersurfaces discovered by E. Brieskorn and exposi-
ted by Milnor [M2]. An interesting relation between the classical
resultant of two polynomials and the coupling invariant first introduced
in Section Three is exploited to give condition for the existence of
torsion in the cyclotomic subgroup, C (Z) . The fibered knots also
give rise to endomorphisms which are also invertible. (This is an
equivalence in dimensions greater than one.) We show that this subgroup
is strictly larger than the cyclotomic subgroup. In the fibered knot
case, the associated isometry s is geometrically given by the monodromy
of a knot, the generating automorphism the cover of the complement of a
knot induced by the normal circle. However, this monodromy is noV always
integral and this is the algebraic reason for the introduction for the
concept of isometric structure.
The final section of the paper is devoted to several geometric
applications of the algebraic computations. We first explore the moti-
vating problem of knot concordance and interpret the localization
sequence geometrically. Then, in Theorem 6.4, we interpret the coupling
invariants as obstructions to splitting knots (up to concordance) as
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