Alexander Polynomials of Ribbon Links

We give a simple argument to show that every polynomial f(t) ∈ Z[t] such that f(1) = 1 is the Alexander polynomial of some ribbon 2-knot whose group is a 1-relator group, and we extend this result to links. It is well known that every Laurent polynomial f(t) ∈ Λ = Z[t, t] with f(1) = 1 is the Alexander polynomial of some ribbon 2-knot [7]. (See also [1, 2], for the fibred case, and §7H of [11], for a construction of knot polynomials by surgery.) We shall give another argument, which seems particularly simple, and which gives a slightly stronger result. We shall then extend this result to higher-dimensional links. In higher dimensions the term “Alexander polynomial” is potentially ambiguous. Let X = S − intK ×D be the knot exterior, π = πK = π1(X) the knot group and X ′ the maximal abelian covering space of X . The homology groups Hq(X ;Z) are finitely generated torsion Λ-modules under the action of Aut(X /X) ∼= Z. They each have a sequence of “Alexander polynomial” invariants ∆qi (K) such that ∆ q i (K) divides ∆qi+1(K) in Λ [8]. Poincaré duality implies that ∆ n+1−q i (K) = ∆ q i (K) for q ≤ [ 2 ], where the overbar is the involution defined by inverting the generators ti. More generally, if L is a μ-component n-link there are similar invariants in Λμ = Z[t ± 1 , . . . , t ± μ ]. In this paper “Alexander polynomial” shall mean the greatest common divisor ∆(π) of the first nonzero elementary ideal of the “Alexander module” A(π) of π. A presentation for this module may be derived from a presentation for π by the free differential calculus. If n > 1 the module has rank μ and ∆(L) = ∆μ(L), but when n = 1 it has rank ≤ μ, with equality if L is concordant to a boundary link. (See [6] for more on Alexander modules.) Let ε : Λμ → Z be the augmentation homomorphism defined by ε(ti) = 1 for all i. Then ε(∆(π)) = 1, since π/π ∼= Z. The burden of this note is that this is the only constraint on such link polynomials, if n > 1. The case n = 2 is of particular interest, for then H1(X ;Z) and duality determine the other homology modules. (When n = 1 and L is a boundary link we must also have ∆ = ∆; there is as yet no such characterization for other classical links.)