Metabelian representations of knot groups

Abstract

The question of determining which finite metabelian groups may be the homomorphic image of a given knot group G is considered in this paper. As a starting point, it is shown that a homomorphism of a knot group onto a metabelian group H such that [H: H′]=n must factor through Zn⦰An9 where Anis the homology group of the n-fold cyclic covering space. This is similar to a theorem of Burde [1 Satz 4], and Reyner [5] has also proven a similar result, showing in effect that such a homomorphism must factor through Z⦰ An. Now, An can be given the structure of a module over the ring Z⟨t⟩ of L-polynomials, and the problem of determining the metabelian factor groups of G can be reduced to determining the factor modules of An. © 1979, University of California, Berkeley. All Rights Reserved.