  Disclinations lines play a key role in many physical processes, from the
fracture of materials to the formation of the early universe. Achieving
versatile control over disclinations is key to developing novel electro-optical
devices, programmable origami, directed colloidal assembly, and controlling
active matter. Here, we introduce a theoretical framework to tailor
three-dimensional disclination architecture in nematic liquid crystals
experimentally. We produce quantitative predictions for the connectivity and
shape of disclination lines found in nematics confined between two thinly
spaced glass substrates with strong planar anchoring. By drawing an analogy
between nematic liquid crystals and magnetostatics, we find that: i)
disclination lines connect defects with the same topological charge on opposite
surfaces, and ii) disclination lines are attracted to regions of the highest
twist. Using polarized light to pattern the in-plane alignment of liquid
crystal molecules, we test these predictions experimentally and identify
critical parameters that tune the disclination lines' curvature. We verify our
predictions with computer simulations and find non-dimensional parameters
enabling us to match experiments and simulations at different length scales.
Our work provides a powerful method to understand and practically control
defect lines in nematic liquid crystals.
