Twisted Alexander polynomials detect the unknot

The group of a nontrivial knot admits a finite permutation representation such that the corresponding twisted Alexander polynomial is not a unit.Comment: This is the version published by Algebraic & Geometric Topology on 14 November 200

Mahler measure, links and homology growth

Let l be a link of d components. For every finite-index lattice in Z^d there is an associated finite abelian cover of S^3 branched over l. We show that the order of the torsion subgroup of the first homology of these covers has exponential growth rate equal to the logarithmic Mahler measure of the Alexander polynomial of l, provided this polynomial is nonzero. Our proof uses a theorem of Lind, Schmidt and Ward on the growth rate of connected components of periodic points for algebraic Z^d-actions.Comment: 13 pages, figures. Small corrections, references updated. To appear in Topolog