Volume bounds for hyperbolic rod complements in the 3-torus

TL;DR

This paper establishes volume bounds for hyperbolic rod complements in the 3-torus, linking geometric properties to rod parameters and improving bounds for specific families.

Contribution

It provides new upper and lower volume bounds for hyperbolic rod complements, utilizing continued fraction parameters, and introduces sharper bounds for certain families.

Findings

Bounds depend on rod parameters and can be loose.

New asymptotically sharp bounds are introduced.

Bounds are expressed via continued fractions.

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.