Knots Arising From the Planar Equal-Mass Collisionless Three-Body Problem

Abstract

We present a series of steps that transform equal-mass collisionless planar three-body configurations into closed curves in ℝ³. Exploiting a centre-of-mass constraint and a size constraint, we reduce the original configuration in ℝ⁶ to ℝ⁴. We then use a rotation and an additional normalisation to place these configurations on the 3-sphere S³, followed by a stereographic projection onto ℝ³. This procedure turns a periodic orbit into a closed curve that can be classified as a knot. We apply our method to the fifteen periodic configurations presented by Šuvakov and Dmitrašinović and show that most of these trajectories become the unknot under our projection. However, three orbits become trefoils and one becomes a figure-eight. We discuss possible directions for future work.