  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.
