Mathematics > Geometric Topology
[Submitted on 14 May 2018 (v1), last revised 14 Nov 2018 (this version, v2)]
Title:Mahler Measure and the Vol-Det Conjecture
View PDFAbstract: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.
Submission history
From: Ilya S. Kofman [view email][v1] Mon, 14 May 2018 22:59:09 UTC (652 KB)
[v2] Wed, 14 Nov 2018 22:06:50 UTC (654 KB)
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