2017/02/21 by Oliver Bräunling, Braunling, Oliver
Mathematics · #Geometric and Algebraic Topology #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1702.06243
We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's theorem on periodic homology. Our results generalize to other rank 1 growth phenomena, e.g. Reidemeister-Franz torsion growth for higher-dimensional knots. We also analyze the exceptional cases where the meromorphic continuation does not exist.