Braiding Dynamics in Active Nematics

In active matter systems, energy consumed at the small scale by individual agents gives rise to emergent flows at large scales. For 2D active nematic microtubule systems, these flows are largely characterized by the dynamics of mobile defects in the nematic director field. As these defects wind about each other, their trajectories trace out braids, and the topological properties of these braids encode the most important global features of the flow. In particular, the topological entropy of a braid quantifies how chaotic the associated flow is. Since microtubule bundles, an extensile system, stretch out exponentially in time, the resultant defect movement must correspond to braids with positive topological entropy. Indeed, we conjecture that the emergent defect dynamics are optimal in that they give braids which maximize the, suitably normalized, topological entropy. In addition to the cases where the active nematic material is confined to a channel or the surface of a sphere, where there is good evidence for our conjecture, we share new experimental data concerning the behavior of microtubules confined to the region between a lattice of pillars.