Understanding and visualizing complex topological defects in 3D active and passive nematics

While much can be understood about two-dimensional nematics by treating point disclinations as quasiparticles, the situation in three dimensions is complicated by the geometrical complexities of defect lines and by the subtleties of "topological charge" in 3D. This talk will present the theory and application of one family of approaches to extracting the important geometrical and topological information about nematic defects from orientation data. The main object of interest is the "rotation vector" or Frank vector at each point on a disclination, which emerges as an eigenvector of two distinct tensors connected with saddle-splay and twist distortions, respectively. Furthermore, along with an orientational degree of freedom, the rotation vector determines the topological charge of closed-loop disclinations. I will review successful applications of these approaches to experimental and simulated nematic orientation data, along with insights provided by these results. I will also discuss a coupling between disclination winding and the Gaussian curvature of topographically patterned confining surfaces.