Mahler Measure and the Vol-Det Conjecture

TL;DR

This paper proves the Vol-Det Conjecture for many alternating links by computing Mahler measures of associated polynomials and introduces a new conjecture relating Mahler measures to hyperbolic volumes.

Contribution

It establishes the Vol-Det Conjecture for numerous families of links and proposes a new lower bound conjecture connecting Mahler measures and hyperbolic volumes.

Findings

Proved the Vol-Det Conjecture for several infinite link families.

Formulated a new lower bound conjecture for Mahler measures.

Confirmed the conjecture for six specific toroidal links.

Abstract

The Vol-Det Conjecture relates the volume and the determinant of a hyperbolic alternating link in $S^3$. We use exact computations of Mahler measures of two-variable polynomials to prove the Vol-Det Conjecture for many infinite families of alternating links. We conjecture a new lower bound for the Mahler measure of certain two-variable polynomials in terms of volumes of hyperbolic regular ideal bipyramids. Associating each polynomial to a toroidal link using the toroidal dimer model, we show that every polynomial which satisfies this conjecture with a strict inequality gives rise to many infinite families of alternating links satisfying the Vol-Det Conjecture. We prove this new conjecture for six toroidal links by rigorously computing the Mahler measures of their two-variable polynomials.