Quaternions and hybrid nematic disclinations

Disclination lines in nematic liquid crystals can exist in different geometric conformations, characterized by their director profile. In certain confined colloidal suspensions and even more prominently in chiral nematics, the director profile may vary along the disclination line. We construct a robust geometric decomposition of director profile in closed disclination loops and use it to apply topological classification to linked loops with arbitrary variation of the profile, generalizing the self-linking number description of disclination loops with the winding number -1/2. The description bridges the gap between the known abstract classification scheme derived from homotopy theory and the observable local features of disclinations, allowing application of said theory to structures that occur in practice.