Volume bounds for hyperbolic rod complements in the 3-torus

Norman Do, Connie On Yu Hui and Jessica S. Purcell

Vol. 339 (2025), No. 1, 167–189

DOI: 10.2140/pjm.2025.339.167

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.

Recently, Hui classified 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.

Keywords

rod complement, link complement, 3-torus, $n$-torus, volume, volume bound, hyperbolic geometry, continued fraction, link complement, Dehn filling, nested annular Dehn filling, surgery, parent manifold, asymptotically sharp, generalised Bézout's lemma

Mathematical Subject Classification

Primary: 57K32

Secondary: 57K10, 57K35, 57Z15

Milestones

Received: 14 January 2025

Revised: 3 August 2025

Accepted: 11 August 2025

Published: 1 September 2025