Lissajous 3-braids

TL;DR

This paper classifies and parametrizes 3-braids generated by collision-free choreographic motions on Lissajous curves, linking them to pseudo-Anosov mapping classes and exploring their geometric and algebraic properties.

Contribution

It introduces a classification and parametrization of Lissajous 3-braids, connecting them to mapping class groups, frieze patterns, and continued fractions.

Findings

Lissajous 3-braids correspond to pseudo-Anosov mapping classes.

Dilatation increases with level and decreases with slope in the Stern-Brocot tree.

Frieze patterns encode geodesic cutting sequences related to quadratic surds.

Abstract

We classify 3-braids arising from collision-free choreographic motions of 3 bodies on Lissajous plane curves, and present a parametrization in terms of levels and (Christoffel) slopes. Each of these Lissajous 3-braids represents a pseudo-Anosov mapping class whose dilatation increases when the level ascends in the natural numbers or when the slope descends in the Stern-Brocot tree. We also discuss 4-symbol frieze patterns that encode cutting sequences of geodesics along the Farey tessellation in relation to odd continued fractions of quadratic surds for the Lissajous 3-braids.