Arithmeticity and commensurability of links in thickened surfaces

TL;DR

This paper classifies right-angled tiling links in thickened surfaces based on their arithmetic properties and pairwise commensurability, using geometric, combinatorial, and number-theoretic methods.

Contribution

It provides a complete characterization of which such links are arithmetic and which are pairwise commensurable, advancing understanding of their geometric and algebraic structures.

Findings

Identifies conditions for arithmeticity of tiling links.

Classifies pairwise commensurability among these links.

Uses symmetry, Coxeter polyhedra, and trace fields for classification.

Abstract

The family of right-angled tiling links consists of links built from regular 4-valent tilings of constant-curvature surfaces that contain one or two types of tiles. The complements of these links admit complete hyperbolic structures and contain two totally geodesic checkerboard surfaces that meet at right angles. In this paper, we give a complete characterization of which right-angled tiling links are arithmetic, and which are pairwise commensurable. The arithmeticity classification exploits symmetry arguments and the combinatorial geometry of Coxeter polyhedra. The commensurability classification relies on identifying the canonical decompositions of the link complements, in addition to number-theoretic data from invariant trace fields.