Curvature-guided topology and self-assembly in chiral nematics and liquid-crystal colloids

Abstract

In soft condensed matter, curvature does more than simply distort an ordered medium: it helps select defect structures, redistribute elastic stress, bias chirality, and guide self-assembly. This review examines how curved, multiply connected, and knotted boundaries in liquid-crystal colloids and confined nematics generate topological defects and localized solitonic textures, and how these structures mediate interactions between mesoscale building blocks. We introduce a unifying framework based on genus, Euler characteristic, anchoring, and chirality, and use it to discuss spherical, handlebody, and boundary-bearing colloids, together with droplets and polymer-dispersed nematics of nontrivial topology. Particular emphasis is placed on the interplay of geometry and topology in determining boojums, disclination loops, hedgehog charges, and linked and knotted defect structures. We then turn to chiral systems hosting skyrmions, torons, hopfions, and related localized textures, highlighting how chirality and confinement stabilize three-dimensional topological states. Finally, we discuss how these concepts translate into design principles for controlled self-assembly, templating, and functional composite materials. More broadly, we argue that liquid-crystal colloids and confined nematics provide experimentally accessible model systems in which curvature, topology, and chirality can be harnessed as programmable tools for designing organized soft matter.