Topological Mixing and Braiding Universality in Polar Active Matter
Connecting the autonomous kinematics of active matter to its emergent macroscopic transport is constrained by the requirement for high-resolution Eulerian velocity fields. Here, we use confined bacterial suspensions as a model active fluid, mapping the sparse Lagrangian trajectories of fluorescent spy cells into (2+1)-dimensional geometric braids that directly encode the spatiotemporal entanglement of the flow. We use the finite-time braiding exponent (FTBE) as a proxy to quantify the topological entropy and chaotic mixing of confined bacterial suspensions. In moderately confined wet systems, we find that hydrodynamic coupling drives a structural transition from a dilute active gas to coherent vortices, and ultimately to active turbulence, revealing three distinct density-dependent regimes of topological mixing. Conversely, truncating the hydrodynamic screening length via extreme confinement drives the system toward a dry active matter limit. In this limit, dense in-plane steric collisions suppress irreducible entanglement and substantially reduce the FTBE at high particle densities. By evaluating the topological complexity generated per encounter, we reveal a transition from a discrete geometric encounter regime to an areal escape mechanism. Finally, we establish a square-root scaling between the FTBE and the effective diffusivity, placing the self-sustained mixing of active fluids into the pathline braiding universality class.