Relationship between topology and geometry of disclination lines in nematics

Investigate the relationship between topological properties of disclination lines in nematic liquid crystals and their geometric realization, clarifying how topological invariants (such as self-linking defined via dividing curves) correspond to geometric features (e.g., twisting and writhing) of the lines.

Background

The paper develops a topological framework using dividing curves and surgery theory to assign orientations and self-linking numbers to disclination lines, and connects these notions to existing geometric treatments (e.g., the ribbon model and Peach–Koehler forces).

While some correspondences between topology and geometry are known (e.g., self-linking manifesting in twisting/writhing in colloidal systems), the authors explicitly note that many aspects of this relationship remain unresolved and call for further investigation.