  This work examines self-mixing in active nematics, a class of fluids in which
mobile topological defects drive chaotic flows in a system comprised of
biological filaments and molecular motors. We present experiments that
demonstrate how geometrical confinement can influence the braiding dynamics of
the defects. Notably, we show that confinement in cardioid-shaped wells leads
to realization of the golden braid, a maximally efficient mixing state of
exactly three defects with no defect creation or annihilation. We characterize
the golden braid state using different measures of topological entropy and the
Lyapunov exponent. In particular, topological entropy measured from the
stretching rate of material lines agrees well with an analytical computation
from braid theory. Increasing the size of the confining cardioid produces a
transition from the golden braid, to the fully chaotic active turbulent state.
