  Active nematics are an important new paradigm in soft condensed matter
systems. They consist of rod-like components with an internal driving force
pushing them out of equilibrium. The resulting fluid motion exhibits chaotic
advection, in which a small patch of fluid is stretched exponentially in
length. Using simulation, this Letter shows that this system can exhibit stable
periodic motion when sufficiently confined to a square with periodic boundary
conditions. Moreover, employing tools from braid theory, we show that this
motion is maximally mixing, in that it optimizes the (dimensionless)
``topological entropy'' -- the exponential stretching rate of a material line
advected by the fluid. That is, this periodic motion of the defects,
counterintuitively, produces more chaotic mixing than chaotic motion of the
defects. We also explore the stability of the periodic state. Importantly, we
show how to stabilize this orbit into a larger periodic tiling, a critical
necessity for it to be seen in future experiments.
