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If a knot K has Alexander
polynomial different from 1, then its knot group, G maps onto some metacyclic
group, Zr Zp. We show that in that case, it also has a homomorphism onto a split
extension of a free abelian group of rank p− 1 by ZrZp, and hence also onto a split
extension of a direct sum of p− 1 cyclic groups of order s by the metacyclic group. In
many cases, (such as if s is coprime with p), this group can be specified
exactly. Otherwise there are a finite number of possibilities. A special case is
Perko’s result that a homomorphism of a knot group onto S3 = Z2Z3 lifts to
S4 = Z2Z3(Z2 ⊕ Z2).
As an application we obtain information about the derived series of G.
In a final section it is shown how to associate a rational polynomial invariant to
every metacyclic representation.
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