Approximations to the volume of hyperbolic knots

TL;DR

This paper explores the relationships between the volume of hyperbolic knots and various algebraic invariants such as the determinant and Mahler measures of Alexander polynomials, supported by computational data and heuristic reasoning.

Contribution

It introduces new conjectural correlations between hyperbolic knot volume and algebraic invariants, supported by computational evidence and heuristic arguments.

Findings

Volume correlates with the determinant of the knot.

Volume correlates with the Mahler measure of the Alexander polynomial.

Volume correlates with the Mahler measure of the twisted Alexander polynomial.

Abstract

We present computational data and heuristic arguments which suggest that given a hyperbolic knot the volume correlates with its determinant, the Mahler measure of its Alexander polynomial and the Mahler measure of the twisted Alexander polynomial corresponding to the discrete and faithful SL(2,C)-representation.