2019/12/30 by Léo Bénard, Jérôme Dubois, Bénard, Léo +5
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.1912.12946
51 pages, comments welcome
arxiv created 2019/12/30 · arxiv updated 2020/01/01
For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we prove the asymptotic behavior for cusped hyperbolic manifolds of finite volume. The proof relies on results of Müller, and Menal-Ferrer and the last author. Using the uniformity of the convergence, we also deduce a similar asymptotic result for the Mahler measures of those polynomials.