Abstract: 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 more and more interesting defect textures, including controllably 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). I will discuss recent work with Gareth Alexander (Warwick) applying the Thom construction from algebraic topology which allows us 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.