Geometry of biperiodic alternating links

TL;DR

This paper investigates semi-regular biperiodic alternating links, determining their hyperbolic structures, volumes, and relationships to Euclidean tilings, and explores their geometric and arithmetic properties including volume bounds and the Volume Density Conjecture.

Contribution

It introduces a method to compute hyperbolic structures and volumes of semi-regular links directly from Euclidean tilings, linking geometric, arithmetic, and topological properties.

Findings

Exact volumes of semi-regular links are determined.

Only two semi-regular links have totally geodesic checkerboard surfaces.

Conditions are given for hyperbolicity and geometric triangulations of biperiodic 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.