Topological classification of 3D periodic orbits in the general three-body problem

Identify a rigorous topological classification scheme for three-dimensional periodic orbits in the Newtonian general three-body problem with finite masses, analogous in purpose to the braid-group and shape-space sphere classifications used for planar periodic orbits, so that such 3D orbits can be systematically characterized up to topological equivalence.

Background

For planar periodic orbits of the Newtonian three-body problem, established topological classification frameworks exist, notably through braid groups and the shape-space sphere. These frameworks enable systematic categorization of distinct planar orbit families.

In contrast, for three-dimensional periodic orbits with finite masses, no accepted topological classification method is known. Given the discovery in this work of 10,059 new 3D periodic orbits—including choreographic and "piano-trio" types—a robust classification scheme would be valuable for organizing, comparing, and analyzing these orbits and for advancing understanding of three-body dynamics in three dimensions.