The main volume–Mahler measure conjecture for biperiodic alternating links

Let $\mathcal{L}$ be any biperiodic alternating link, with toroidally alternating $\Lambda$-quotient link $L$. Let $p(z,w)$ be the characteristic polynomial of the toroidal dimer model on $G_{\mathcal{L}}^b$. Main conjecture.

$$\operatorname{vol}^{\lozenge}(L) \leq 2\pi\,\mathrm{m}(p(z,w)).$$

This conjecture relates the volume density of a biperiodic alternating link to the Mahler measure of its dimer-model characteristic polynomial. The paper proves it for six examples; it remains open in general.