Braiding Dynamics in Active Nematic Flows

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 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 experimental data and simulations. Additionally, we will share some predictions for the behavior of microtubules confined to the region between a lattice of pillars, based on recent advances in braid theory.