Lifting group homomorphisms

Richard I. Hartley

Vol. 105 (1983), No. 2, 311–320

DOI: 10.2140/pjm.1983.105.311

Abstract

If a knot K has Alexander polynomial different from 1, then its knot group, G maps onto some metacyclic group, Z_rZ_p. 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 Z_rZ_p, 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.

Mathematical Subject Classification 2000

Primary: 57M25

Milestones

Received: 24 November 1980

Published: 1 April 1983