Volume bounds for hyperbolic rod complements in the 3-torus
Abstract
The study of rod complements is motivated by rod packing structures in crystallography. We view them as complements of links comprised of Euclidean geodesics in the 3-torus. Recent work of the second author classifies when such rod complements admit hyperbolic structures, but their geometric properties are yet to be well understood. In this paper, we provide upper and lower bounds for the volumes of all hyperbolic rod complements in terms of rod parameters, and show that these bounds may be loose in general. We introduce better and asymptotically sharp volume bounds for a family of rod complements. The bounds depend only on the lengths of the continued fractions formed from the rod parameters.
Cite
@article{arxiv.2409.02357,
title = {Volume bounds for hyperbolic rod complements in the 3-torus},
author = {Norman Do and Connie On Yu Hui and Jessica S. Purcell},
journal= {arXiv preprint arXiv:2409.02357},
year = {2025}
}
Comments
20 pages, 5 figures. V3: Added a figure and made minor changes based on reviewer's comments