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Simulated optical polarizing microscopy (OPM) textures can be used to help compare a director ansatz with an associated experimental OPM texture. Here, we present a pedagogical treatmement of a simulation formalism based on the method of Jones matrices. We focus on toroidal geometries as an example and perform computer simulations of the nematic textures seen between crossed-polarizers. For tangentially-anchored nematic tori, we compare qualitative and quantitative features between experimental and simulated OPM textures and discuss the influence of twist in the director field. In addition, we quantify the effect of averaging over multiple wavelengths, the effect of voxel size, how refraction affects the comparison between experimental and simulated OPM textures, and the effect of deviations from the toroidal shape. For homeotropically-anchored nematic tori, we adapt the classic escaped-radial director field found in homeotropically-anchored capillaries to a toroidal geometry to produce simulated OPM textures. Lastly, we show that the same signatures of twist in the tangentially-anchored nematic toroids persist in the homeotropically-anchored nematic toroids. [ABSTRACT FROM AUTHOR]

Due to the increased interest in directly visualising partially-ordered fluids, there is a need for techniques that enable determining the associated director field from intensity images in robust and precise ways. Coherence-enhanced diffusion filtering (CEDF) is a computer vision technique designed to determine the local direction along which the intensity fluctuates the least. We pedagogically illustrate how CEDF can be applied to such partially-ordered systems using a microtubule-based active nematic as an example. In this system, the local microtubule orientation is visible using fluoresence microscopy. We illustrate how to choose parameters in the analysis, and demonstrate the technique is robust by applying it to data corresponding to an active nematic in flat space and an active nematic confined to the surface of a torus. Lastly, we show how to leverage topology to calculate the error in the analysis and find that in the examples presented, the influence of errors is negligible. [ABSTRACT FROM AUTHOR]

We stabilize nematic droplets with handles against surface tension-driven instabilities, using a yield-stress material as outer fluid, and study the complex nematic textures and defect structures that result from the competition between topological constraints and the elasticity of the nematic liquid crystal. We uncover a surprisingly persistent twisted configuration of the nematic director inside the droplets when tangential anchoring is established at their boundaries, which we explain after considering the influence of saddle splay on the elastic free energy. For toroidal droplets, we find that the saddle-splay energy screens the twisting energy, resulting in a spontaneous breaking of mirror symmetry; the chiral twisted state persists for aspect ratios as large as ~20. For droplets with additional handles, we observe in experiments and computer simulations that there are two additional -1 surface defects per handle; these are located in regions with local saddle geometry to minimize the nematic distortions and hence the corresponding elastic free energy. [ABSTRACT FROM AUTHOR]