The Two Eyes Lemma: a linking problem for horoball necklaces

TL;DR

This paper investigates the geometric linking problem of horoball necklaces in hyperbolic 3-space, establishing conditions on bead diameters and tangencies, and identifying a continuous family of configurations.

Contribution

It introduces a new linking problem for horoball necklaces, providing a detailed analysis of configurations and proving constraints on bead sizes and tangencies.

Findings

All beads in the 8-bead necklace must have diameter one.

The two linked spheres are tangent to each other.

There exists a continuous family of distinct configurations.

Abstract

In the course of our work on low-volume hyperbolic 3-manifolds, we came upon a linking problem for horoball necklaces in $\mathbb{H}^3$. A horoball necklace is a collection of sequentially tangent beards (i.e. spheres) with disjoint interiors lying on a flat table (i.e. a plane) such that each bead is of diameter at most one and is tangent to the table. In this note, we analyze the possible configurations of an 8-bead necklace linking around two other diameter-one spheres on the table. We show that all the beads are forced to have diameter one, the two linked spheres are tangent, and that each bead must kiss (i.e. be tangent to) at least one of the two linked spheres. In fact, there is a 1-parameter family of distinct configurations.