  The particle-scale dynamics of granular materials have commonly been
characterized by the self-diffusion coefficient $D$. However, this measure
discards the collective and topological information known to be an important
characteristic of particle trajectories in dense systems. Direct measurement of
the entanglement of particle space-time trajectories can be obtained via the
topological braid entropy $\Sbraid$, which has previously been used to quantify
mixing efficiency in fluid systems. Here, we investigate the utility of
$\Sbraid$ in characterizing the dynamics of a dense, driven granular material
at packing densities near the static jamming point $\phi_J$. From particle
trajectories measured within a two-dimensional granular material, we typically
observe that $\Sbraid$ is well-defined and extensive. However, for systems
where $\phi \gtrsim 0.79$, we find that $\Sbraid$ (like $D$) is not
well-defined, signifying that these systems are not ergodic on the experimental
timescale. Both $\Sbraid$ and $D$ decrease with either increasing packing
density or confining pressure, independent of the applied boundary condition.
The related braiding factor provides a means to identify multi-particle
phenomena such as collective rearrangements. We discuss possible uses for this
measure in characterizing granular systems.
