2008/01/14 by Daniel S. Silver, Susan G. Williams, Silver, Daniel S. +1
Mathematics · #37B40 #57M25 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #math.DS #math.GT #msc:37B40 #msc:57M25
paper · pdf · doi:10.48550/arxiv.0801.2118
This version contains corrections and improvements in exposition. 38 pages, 4 figures
arxiv created 2009/04/30 · arxiv updated 2009/12/01
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.