since the boundary of V is a homotopy sphere. Orienting V so that
K is the oriented boundary of V determines a unique positive normal
direction to V in M . The Seifert linking form L on the Seifert
manifold V is defined on M = H- .
by L(x,y) is the
linking number of x with the homology class obtained by pushing y a
small distance in the positive normal direction. Since the intersection
pairing , is unimodular there is an endomorphism t of M
(related to the monodromy of the knot) defined by the relation t(x),y =
L(x,y) . This relation satisfies the relation
tx,y + x,ty = x,y
The triple (M,,,t) defines an integral isometric structure associated
to the knot (M/K) . These structures from a monoid under the operation
of orthogonal direct sum. Furthermore, one can show that (M,K) is
concordant to the standard knot (S , S ) where the embedding is the
standard inclusion, if and only if there is a t invariant submodule H
of one-half the rank of M and on which the form is identically zero.
These are the metabolic isometric structures and C (S) is the quotient
monoid obtained by dividing out the metabolic structures. In fact,
C {%) is a group and the group structures mimics the geometric group
structure on the concordance classes of spherical knot defined by the
appropriate connected sum. Another isomorphic obstruction group for
this problem has been defined by Cappell and Shaneson [CS] which can be
used to study the codimension two embedding problem for arbitrary mani-
folds. Our computations, however, will employ Levine s formulation
which has the advantage of defining an a priori obstruction without the
need for the completion of surgery up to the middle dimension (Problem 5
in Shaneson [Sh]). The setting also applies to the study of cyclic
symmetries of spherical knots as explicated in [St].
The algebraic computations of this paper proceed on two levels. The
first level is concerned with the existence of a self dual invariant
integral lattice (a finitely generated abelian subgroup of maximal rank)
representing a given element of the rational knot concordance group. A
necessary and sufficient condition for the existence of a lattice
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