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

Twisted Alexander Invariants of Twisted Links

Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be the d-component link in a homology 3-sphere that results from performing 1/q-surgery on the last component. Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtained. The behavior of the invariants as q increases without bound is described.Comment: 21 pages, 6 figure

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

Twisted Alexander Polynomials and Representation Shifts

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group.Comment: 7 pages, no figure