Summary: We study the streamlines of the velocity field generated by two Helmholtz vortex rings which are fixed in space. Chaos can arise directly from a perturbation of periodic orbits. Trajectories are regular far away from linked vortex rings, and chaotic far away from unlinked vortex rings. Intersections of vortex rings produce intricate streamline topologies. The correspondence between the topology of the streamlines and the topology of the vortex rings is discussed.