Seeing and Sculpting Nematic Liquid Crystal Textures with the Thom construction

Nematic liquid crystals are the foundation for modern display technology and also exhibit topological defects that can readily be seen under a microscope. Recently, experimentalists have been able to create and control several new families of interesting defect textures, including reconfigurably knotted defect lines around colloids (Ljubljana) and the ``toron,'' a pair of hedgehogs bound together with a ring of double-twist between them (CU Boulder). We apply the Thom construction from algebraic topology to visualize 3 dimensional molecular orientation fields as certain colored surfaces in the sample. These surfaces turn out to be a generalization to 3 dimensions of the dark brushes seen in Schlieren textures of two-dimensional samples of nematics. Manipulations of these surfaces correspond to deformations of the nematic orientation fields, giving a hands-on way to classify liquid crystal textures which is also easily computable from data and robust to noise.