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 (ANMT) 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. We introduce a minimal model of ANMT systems based on the topological properties of these braids. In particular, we consider the topological entropy of braids, which quantifies how chaotic the associated flow must be. 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 often optimal in that they give braids which maximize the, suitably normalized, topological entropy. We will look at the dynamics of four +1/2 defects on a sphere as a case study, using both simulations and a reinterpretation of experimental data from the literature.