Three-dimensional orientational order in systems whose ground states possess non-zero, chiral gradients typically exhibits line-like structures or defects: $\lambda$ lines in cholesterics or Skyrmion tubes in ferromagnets for example. Here we show that such lines can be identified as a set of natural geometric singularities in a unit vector field, the generalisation of the umbilic points of a surface. We characterise these lines in terms of the natural vector bundles that the order defines and show that they give a way to localise and identify Skyrmion distortions in chiral materials -- in particular that they supply a natural representative of the Poincar\'{e} dual of the cocycle describing the topology. Their global structure leads to the definition of a self-linking number and helicity integral which relates the linking of umbilic lines to the Hopf invariant of the texture.Comment: 14 pages, 9 figure

We show that the number of distinct topological states associated with a given knotted defect, L, in a nematic liquid crystal is equal to the determinant of the link L. We give an interpretation of these states, demonstrate how they may be identified in experiments, and describe the consequences for material behavior and interactions between multiple knots. We show that stable knots can be created in a bulk cholesteric and illustrate the topology by classifying a simulated Hopf link. In addition, we give a topological heuristic for the resolution of strand crossings in defect coarsening processes which allows us to distinguish topological classes of a given link and to make predictions about defect crossings in nematic liquid crystals.

Knots and knotted fields enrich physical phenomena ranging from DNA and molecular chemistry to the vortices of fluid flows and textures of ordered media. Liquid crystals provide an ideal setting for exploring such topological phenomena through control of their characteristic defects. The use of colloids in generating defects and knotted configurations in liquid crystals has been demonstrated for spherical and toroidal particles and shows promise for the development of novel photonic devices. Extending this existing work, we describe the full topological implications of colloids representing nonorientable surfaces and use it to construct torus knots and links of type (p,2) around multiply twisted Möbius strips.topological defects | homotopy theory | metamaterials C ontrolling and designing complex 3D textures in ordered media is central to the development of advanced materials, photonic crystals, tunable devices or sensors, and metamaterials (1-10), as well as to furthering our basic understanding of mesophases (11-13). Topological concepts, in particular, have come to play an increasingly significant role in characterizing materials across a diverse range of topics from helicity in fluid flows (14, 15) and transitions in soap films (16) to molecular chemistry (17), knots in DNA (18), defects in ordered media (19, 20), quantum computation (21, 22), and topological insulators (23). Topological properties are robust, because they are protected against all continuous deformations, and yet flexible for the same reason, allowing for tunability without loss of functionality.Some of the most intricate and interesting textures in ordered media involve knots. Originating with Lord Kelvin's celebrated "vortex atom" theory (24), the idea of encoding knotted structures in continuous fields has continued in magnetohydrodynamics (25), fluid dynamics (15), high-energy physics (26-28), and electromagnetic fields (29, 30), and has seen recent experimental realizations in optics (31), liquid crystals (32), and fluid vortices (33). Tying knots in a continuous field involves a much greater level of complexity than in a necktie, or rope, or even a polymer or strand of DNA. In a field, the knot is surrounded by material that has to be precisely configured so as to be compatible with the knotted curve. However, this complexity brings its own benefits, for the full richness of the mathematical theory of knots is naturally expressed in terms of the properties of the knot complement: everything that is not the knot. In this sense, knotted fields are ideally suited to directly incorporate and experimentally realize the full scope of modern knot theory.Liquid crystals are orientationally ordered mesophases, whose unique blend of soft elasticity, optical activity, and fluid nature offers a fertile setting for the development of novel metamaterials and the study of low-dimensional topology in ordered media. Much of the current focus centers on colloidal systems--colloidal particles dispersed in a liquid crystal host--which have a dual charac...

We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in double-struckR3, showing that the distinct homotopy classes have a 1–1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group has representatives that are planar and characterize the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterization. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of τ lines in cholesterics.

We propose a construction of a cholesteric pitch axis for an arbitrary nematic director field as an eigenvalue problem. Our definition leads to a Frenet-Serret description of an orthonormal triad determined by this axis, the director, and the mutually perpendicular direction. With this tool, we are able to compare defect structures in cholesterics, biaxial nematics, and smectics. Though they all have similar ground state manifolds, the defect structures are different and cannot, in general, be translated from one phase to the other.