14 NEAL W. STOLTZFUS

(-l)de9 ff(l-X) .

Proposition 1.11. The minimal polynomial of an isometric structure

is self-dual. Furthermore, the minimal polynomial of a metabolic struc-

ture is of the form cp(x) = f(X)f*(X) for some monic polynomial f .

Proof: The first conclusion was reached in the preceding paragraph.

Now, suppose N is a metabolic submodule of the isometric structure

(M, # #t) . There is the following exact sequence of R[X] modules:

i P

0 - N - M - Hoiri (N,R) - 0

R

where P(m) = (n - m,n) and the R[X]-module structure on Horn (M,R) is

given by (Xh)(n) = h((l-t)n) . It therefore has minimal polynomial

(-1) eg f(1-X) if f is the minimal polynomial of N . By the behav-

ior of the minimal polynomial with respect to short exact sequences the

result follows.

Remarks: i) Let P be the Grothendieck group of the monoid M of

monic polynomials of positive degree in R[X] under polynomial multipli-

cation. The involution defined on M by (f) = f extends to the

2

Grothendieck group and we can form the group P (R) = H (C2;P) which is

easily identified with the quotient group { f in R[X] : f = f }/

{ g: g = hh } . The homomorphism A: C (R) - P (R) induced by

A(M, / #t) = characteristic polynomial of t is then an obstruction to

making an isometric structure metabolic. We will refer to this as the

Alexander invariant.

ii) The characteristic polynomial cp(X) for an integral isometric

structure is related to the classical Alexander polynomial of the corres-

ponding knot by cp(x) = (-€) X D(l-X~ ) , 2h = degree D , where D (X)

is the Alexander polynomial (see [K2]}.

We will denote Witt-equivalence classes of isometric structures with

restrictions on the R[X] - module structure by subscripting our symbol

for the concordance group. Two such restrictions will be needed:

a) Let T be a multiplicative subset of R[X] (i.e. closed under

multiplication). Then C (R) will denote the concordance classes of