Volume function and Mahler measure of exact polynomials

TL;DR

This paper investigates exact polynomials related to knot complements, establishing a connection between their Mahler measure and a volume function, and providing formulas and criteria for their properties.

Contribution

It introduces a volume function approach to compute Mahler measures of exact polynomials and offers new criteria and interpretations related to $A$-polynomials.

Findings

Mahler measure expressed via volume function extrema

Mahler measure divided by π exceeds volume function amplitude

K-theoretical criterion for polynomial factors of $A$-polynomials

Abstract

We study a class of 2-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the 2-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a $K$-theoretical criterium for a polynomial to be a factor of an $A$-polynomial and give a topological interpretation of its Mahler measure.