Geometry of biperiodic alternating links

Abstract

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi-regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field. We show that only two semi-regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links.