Morse Theory and Meron Mediated Interactions Between Disclination Lines in Nematics

The topological understanding of nematic liquid crystals is traditionally centered on singularities, or defects, and their classification via homotopy theory. However, this approach has ultimately proved insufficient to properly capture a range of complex behaviours that have been reported in three dimensions. To address this, we argue that a finer understanding of topology is required, in which non-singular but non-trivial topological solitons - so-called merons - play a central role in mediating interactions between disclination lines. We present a comprehensive framework for capturing such behaviour that draws heavily on aspects of Morse theory; the key notion being that merons appear singular under projection onto a two-dimensional surface. This permits the use of singularity theory and dividing curves to characterise nematic textures via tomography, as well as an understanding of topological transitions via surgery theory. We use our ideas to understand and classify complex three-dimensional behaviours, such as the linking, rewiring and crossing of disclination lines, as well as to provide a new perspective on defect charge.