Higher even dimensional Reidemeister torsion for torus knot exteriors

TL;DR

This paper investigates the asymptotic behavior of higher-dimensional Reidemeister torsion for torus knot exteriors, revealing convergence properties and classifying certain group representations.

Contribution

It introduces new asymptotic results for Reidemeister torsion in torus knot exteriors and classifies SL(2,C)-representations inducing acyclic higher-dimensional representations.

Findings

Logarithm of torsion sequence converges to zero as dimension increases.

Provides classification of SL(2,C)-representations inducing acyclic higher-dimensional representations.

Connects torsion asymptotics with representation theory of torus knot groups.

Abstract

We study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. M\"uller and P. Menal-Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of log |the higher dimensional Reidemeister torsion of a torus knot exterior with SL(2N,C)-representation| / (2N)^2 converges to zero when N goes to infinity. We also give a classification for SL(2,C)-representations of torus knot groups, which induce acyclic SL(2N,C)-representations.