Dynamics of Twisted Alexander Invariants
Abstract
The Pontryagin dual of the twisted Alexander module for a d-component link and GL(N,Z) representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. In the case of a knot, its associated topological entropy is the logarithmic growth rate of the number of torsion elements in the twisted first-homology group of r-fold cyclic covers of the knot complement, as r goes to infinity. Total twisted representations are introduced, and their properties are studied. The twisted 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.
Cite
@article{arxiv.0801.2118,
title = {Dynamics of Twisted Alexander Invariants},
author = {Daniel S. Silver and Susan G. Williams},
journal= {arXiv preprint arXiv:0801.2118},
year = {2009}
}
Comments
This version contains corrections and improvements in exposition. 38 pages, 4 figures