Dynamics of twisted Alexander invariants

The Pontryagin dual of the based Alexander module of a link twisted by a GL N Z representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. Its topological entropy is the exponential growth rate of the number of torsion elements of twisted homology groups of abelian covers of the link exterior. isted Alexander polynomial obtained from any nonabelian parabolic SL 2 C representation of a 2-bridge knot group is seen to be nontrivial. The zeros of any twisted Alexander polynomial of a torus knot corresponding to a parabolic SL 2 C representation or a finite-image permutation representation are shown to be roots of unity.