Bipyramid Volume, Mahler Measure and Some $\mathbb{Z}^2$-periodic Links

TL;DR

This paper investigates the relationship between bipyramid volume and Mahler measure in certain links, confirming a conjecture for specific cases through explicit calculations and connections to lattice models.

Contribution

It confirms a conjecture relating bipyramid volume and Mahler measure for specific links using new computational methods and lattice graph connections.

Findings

Confirmed the conjecture for two examples.

Calculated five additional examples using lattice graph methods.

Established links between dimer models, spanning trees, and link invariants.

Abstract

Champanerkar, Kofman and Lal\'{i}n conjectured an inequality between bipyramid volume of links and Mahler measure of associated dimer models induced from alternating links on torus. Hyperbolic volume and Mahler measure can be related for isoradial graphs, which allows us to confirm the conjecture for two examples. By exploiting a connection between perfect matchings of dimer models and spanning trees on lattices, five more examples are calculated.