Rod packings in the 3-torus and hyperbolic geometry

Abstract

Rod packings are used in crystallography to describe crystal structures with chains of particles, and each rod packing can be viewed as a union of disjoint geodesics in the 3-torus. In a joint work with Jessica Purcell, we used hyperbolic geometry and tools from the 3-sphere to study the complements of geodesic knots and links (also called rods) in the 3-torus.  In this talk, I would like to give motivations for our project, introduce the tools we used, which include a beautiful theorem of Thurston, and share results on geometric classification of rod complements in the 3-torus and their volume estimates.