Point masses moving in 2+1 dimensions draw out braids in space-time. If they move under the influence of some pairwise potential, what braid types are possible? By starting with fictional paths of the desired topology and `relaxing' them by minimizing the action, we explore the braid types of potentials of the form V ~ r^a from a <= -2, where all braid types occur, to a=2 where the system is integrable. We also discuss issues of symmetry and stability, both of the dynamics and of the algorithm. We propose this kind of topological classification as a tool for describing many-body dynamics.