UNRAVELING THE INTEGRAL KNOT CONCORDANCE GROUP 13

Observing that N is invariant, by Proposition 1.4 M 3 (-L /L) is

metabolic and therefore M is Witt equivalent to L /L .

The existence of an anisotropic representative follows since we have

shown that an isometric structure with a non-trivial invariant self-

annihilating subspace is Witt equivalent to a structure with strictly

smaller rank, a process that must eventually terminate in an anisotropic

representative by the finite generation of M .

Corollary 1.9. If L is an invariant pure submodule of an isome-

tric structure (M, , ,t) and L ^ L then M is Witt equivalent to

(L /L, , n»t0) where the form , _ and the endomorphism tn are

induced from their restrictions to L

Corollary 1.10. Every isometric structure is equivalent to a struc-

ture with t injective.

Proof: Let L = kernel t . Then L c L since for x and y in

L , 0 = t(x),y = x,(1-t)(y) = x,y .

For the field K , an isometric structure with t injective is

equivalent to a more familiar object. For t is now an isomorphism and

s = 1 - t is defined. But sx,sy = x,sy - t x,sy = x,x -

x,t y - t x,t y + t x,t y = x,y since by defining relation

ii), t(t_1x) ,t_1y = t_1x, (1-t)(t~1y) . This is the traditional

notion of an isometry of a bilinear form. We further note that the

original structure is metabolic if and only if there is a self-annihila-

ting subspace of one-half the dimension of M invariant under the

isometry s .

By the usual algebraic trick of defining the action of an indeter-

minate X to be the same as that of t , we may view M as a module

over the polynomial ring R[X] . From the identity tx,y = x,(l-t)y

and bilinearity the formula f(t)x,y = x,f(l-t)y holds for any

integral polynomial f in R[X] . By the nonsingularity of the form,

if f(X) annihilates the module M , if and only if f(1-X) does so.

Therefore if CD (X) is the minimal polynomial of t , cp(l-x)| c p (x) and

by comparing the highest coefficients, we have (-1)

g

* cp(l-x) = c p (x) .

For any polynomial f(X) , we may define the dual polynomial f (X) =