Global Defect Topology in Nematic Liquid Crystals

We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in $\mathbb{R}^3$, showing that the distinct homotopy classes have a 1-1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group have representatives that are planar and characterise the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterisation. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of $\tau$ lines in cholesterics.