Year: 2018
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Gabai, David, Meyerhoff, Robert, Yarmola, Andrew
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2801955085