The Many Faces of Lorenz Knots

Abstract

One of the greatest pleasures in doing mathematics (and one of the surest signs of being onto something really relevant) is discovering that two apparently completely unrelated objects actually are one and the same thing. This is what Etienne Ghys, of the Ecole Normale Superieure de Lyon, did a few years ago (see [1] for the technical details), showing that the class of Lorenz knots, pertaining to the theory of chaotic dynamical systems and ordinary differential equations, and the class of modular knots, pertaining to the theory of 2-dimensional lattices and to number theory, coincide. In this short note we shall try to explain what Lorenz and modular knots are, and to give a hint of why they are the same. See also [2] for a more detailed but still accessible presentation, containing the beautiful pictures and animations prepared by Jos Leys [3], a digital artist, to illustrate Ghys’ results.