Knot theory realizations in nematic colloids

Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.From the invention of ropes and textiles, up to the present day, knots have played a prominent role in everyday life, essential crafts, and artistic expression. Beyond the simple tying of strings, the intriguing irreducibility of knots has led to Kelvin’s vortex model of atoms, and, subsequently, a more systematic study of knots and links in the context of knot theory (1–3). As a branch of topology, knot theory is a developing field, with many unresolved questions, including the ongoing search for an algorithm that will provide an exact identification of arbitrary knots.As knots cannot be converted one into another without the crossing of the strands––a discrete singular event––knotting topologically stabilizes the structure. In physical fields, this coexistence of discrete and continuous phenomena leads to the stabilization of geometrically and topologically nontrivial high-energy excitations (4, 5). Examples of strand-like objects in physics that can be knotted include vortices in fluids (6–9), synthetic molecules (10, 11), DNA, polymer strands and proteins (12–14), electromagnetic field lines (15, 16), zero-intensity loci in optical interference patterns (17), wave functions in topological insulators (18), cosmic strings (19), and defects in a broad selection of ordered media (20–23).Nematic liquid crystals (NLC) are liquids with a local apolar orientational order of rod-like molecules. The director field, which describes the spatial variation of the local alignment axis, supports topological point and line defects, making it an interesting medium for the observation of topological phenomena (20, 24). Defect structures in NLC and their colloidal composites (25) have been extensively studied for their potential in self-assembly and light control (26), but also to further the theoretical understanding of topological phenomena in director fields (23, 27). Objects of interest include chiral solitons (28, 29), fields around knotted particles (30–35), and knotted defects in nematic colloids (36–42). Each of these cases is unique, as the rules of knot theory interact with the rules and restrictions of each underlying material and confinement. The investigation of knotted fields is thus a specialized topic where certain theoretical aspects of knot theory emerge in a physical context.In nematic colloids––dispersions of spherical particles confined in a twisted nematic (TN) cell––disclination lines entangle arrays of particles into “nematic braids,” which can be finely controlled by laser tweezers to form various linked and knotted structures (38, 39, 43). In this paper, we focus on the diverse realizations of knot theory in such nematic colloidal structures. We complement and extend the classification and analysis of knotted disclinations from refs. 32, 38, 40 with the direct application of graph and knot theory to polarized optical micrographs. We further analyze the nematic director with constructed Pontryagin–Thom surfaces and polynomial knot invariants, which enables a comprehensive topological characterization of the knotted nematic field based on experimental data and analytical tools. We use a λ-retardation plate to observe and distinguish differently twisted domains in the optical micrograph, which correspond to medial graphs of the represented knots and contribute to the Pontryagin–Thom construction of the nematic director. Finally, we explore the organization of the space of possible configurations on a selected rectangular particle array and discuss the observed hierarchy of entangled and knotted structures.