Les noeuds de Lorenz

TL;DR

This survey explores Lorenz knots, detailing their construction, properties, and a correspondence with modular knots, while introducing new results linking trivial orbits to the class group and isotopy of inverse images.

Contribution

It provides a comprehensive overview of Lorenz knots, proves classical properties, and introduces new results on the relationship between Lorenz flow orbits and algebraic structures.

Findings

Closure of positive braids are fibered knots

Ghys' correspondence links modular and Lorenz knots

Trivial orbits form a subgroup of the class group

Abstract

This article is a survey on Lorenz knots. We describe the original construction, prove several classical properties, in particular the fact that the closure of a positive braid is a fibered knot, and describe Ghys'correspondance between modular knots and Lorenz knots. We also prove two new properties, namely that following Ghys' correspondance, the images of trivial orbits of the Lorenz flow form a subgroup of the class group, and that the reverse images of an element in the class group and of its inverse are isotopic orbits.