Topological tori as abstract art

Abstract

Topologists have known for almost three decades that there are exactly four regular homotopy classes for marked tori immersed in 3D Euclidean space. Instances in different classes cannot be smoothly transformed into one another without introducing tears, creases, or regions of infinite curvature into the torus surface. In this article, a quartet of simple, easy-to-remember representatives for these four classes is depicted to expose this classification to a broader audience. It is shown explicitly how the four models differ from one another, and examples are given of how a more complex immersion of a torus can be transformed into its corresponding representative. These mathematical visualization models are juxtaposed with artistic sculptures employing more classical torus shapes.