Topological defects in nematic and smectic liquid crystals

Liquid crystals are materials with interesting symmetries and order, and their science and applications are a playground of geometry and elasticity. I discuss various aspects of the study of topology and topological defects in liquid crystalline systems beyond the traditional calculation of homotopy groups. I consider two problems. First, how can we better visualize three dimensional nematic orientation fields given that many experiments and simulations now are creating and manipulating configurations with complicated topology? I show that the Pontryagin-Thom construction leads to a natural set of colored surfaces, generalizing the prototypical dark brushes seen in Schlieren textures between crossed polarizers. If we are interested in properties preserved under smooth deformations, these colored surfaces can stand in for the rest of the configuration, leading to a reduction in dimensionality of the data to be considered, as well as an aesthetically pleasing representation of what's happening. The second problem is the relationship between translational order and orientational order in smectic liquid crystal systems. Smectics are liquid crystals which organize themselves into layers where the naive generalization of the calculation of homotopy groups leads to results on defects which are incorrect. I show that these difficulties arise from the lack of independence between translational order and orientational order. These considerations are neatly captured by a surface model of two-dimensional smectics. By looking at a smectic configuration as a graph over the sample space, the nature of the rules governing the defects and their relationship to the symmetries of the system are clarified.