The leading coefficient of the $L^2$-Alexander torsion

TL;DR

This paper establishes bounds on the leading coefficients of the $L^2$-Alexander torsion for 3-manifolds, relates them to hyperbolic volume, and explicitly computes these coefficients for 2-bridge knot exteriors.

Contribution

It provides new bounds and explicit calculations for the leading coefficient of the $L^2$-Alexander torsion, especially for 2-bridge knots.

Findings

Bounds on the leading coefficients in terms of hyperbolic volume

Equality of bounds for certain knot exteriors

Explicit computation for 2-bridge knots

Abstract

We give upper and lower bounds on the leading coefficients of the $L^2$-Alexander torsions of a $3$-manifold $M$ in terms of hyperbolic volumes and of relative $L^2$-torsions of sutured manifolds obtained by cutting $M$ along certain surfaces. We prove that for numerous families of knot exteriors the lower and upper bounds are equal, notably for exteriors of 2-bridge knots. In particular we compute the leading coefficient explicitly for 2-bridge knots.