Mixing of passive scalars in fluids is often effected via moving stirrers. For very viscous flows, efficient motions of the stirring rods have recently been investigated from a topological perspective. The theory of braids is used the to this: the motion of the rods is visualized in a space-time plot, and regarded as a tangle of strings. How tangled these strings are says much about the mixing property of the stirrer motions. Here, I extend this formalism to periodic orbits in addition to rods. I show that some periodic orbits have the net effect of a tiny rod--a ghost rod--helping to stir the fluid.