Smoke, fog, jelly, paints, milk and shaving cream are common everyday examples of colloids 1 , a type of soft matter consisting of tiny particles dispersed in chemically distinct host media. Being abundant in nature, colloids also find increasingly important applications in science and technology, ranging from direct probing of kinetics in crystals and glasses 2 to fabrication of third-generation quantum-dot solar cells 3 . Because naturally occurring colloids have a shape that is typically determined by minimization of interfacial tension (for example, during phase separation) or faceted crystal growth 1 , their surfaces tend to have minimum-area spherical or topologically equivalent shapes such as prisms and irregular grains (all continuously deformable-homeomorphic-to spheres). Although toroidal DNA condensates and vesicles with different numbers of handles can exist 4-7 and soft matter defects can be shaped as rings 8 and knots 9 , the role of particle topology in colloidal systems remains unexplored. Here we fabricate and study colloidal particles with different numbers of handles and genus g ranging from 1 to 5. When introduced into a nematic liquid crystal-a fluid made of rod-like molecules that spontaneously align along the so-called "director" 10 -these particles induce threedimensional director fields and topological defects dictated by colloidal topology. Whereas electric fields, photothermal melting and laser tweezing cause transformations between configurations of particle-induced structures, three-dimensional nonlinear optical imaging reveals that topological charge is conserved and that the total charge of particle-induced defects always obeys predictions of the Gauss-Bonnet and Poincaré-Hopf index theorems [11][12][13] . This allows us to establish and experimentally test the procedure for assignment and summation of topological charges in threedimensional director fields. Our findings lay the groundwork for new applications of colloids and liquid crystals that range from topological memory devices 14 , through new types of self-assembly [15][16][17][18][19][20][21][22][23] , to the experimental study of low-dimensional topology 6,7,[11][12][13] .Although a coffee mug and a doughnut look different to most of us, they are topologically equivalent solid tori or handlebodies of genus g = 1, both being different from, say, balls and solid cylinders of genus g = 0, to which they cannot be smoothly morphed without cutting 11,12 . In a similar way, molecules can form topologically distinct structures including rings, knots and other molecular configurations satisfying the constraints imposed by chemical bonds 24 . Although the topology of shapes, fields and defects is important in many phenomena and in theories ranging from the nature of elementary particles to early-Universe cosmology 25,26 , topological aspects of colloidal systems (composed of particles larger than molecules and atoms but much smaller than the objects that we encounter in our everyday life) are rarely explored. Typically dealing with...

We explore spatially localized solitonic configurations of a director field, generated using optical realignment and laser-induced heating, in frustrated chiral nematic liquid crystals confined between substrates with perpendicular surface anchoring. We demonstrate that, in addition to recently studied torons and Hopf-fibration solitonic structures (hopfions), one can generate a host of other axially symmetric stable and metastable director field configurations where local twist is matched to the surface boundary conditions through introduction of point defects and loops of singular and nonsingular disclinations. The experimentally demonstrated structures include the so-called "baby-skyrmions" in the form of double twist cylinders oriented perpendicular to the confining substrates where their double twist field configuration is matched to the perpendicular boundary conditions by loops of twist disclinations. We also generate complex textures with arbitrarily large skyrmion numbers. A simple back-of-the-envelope theoretical analysis based on free energy considerations and the nonpolar nature of chiral nematics provides insights into the long-term stability and diversity of these inter-related solitonic field configurations, including different types of torons, cholestric-finger loops, two-dimensional skyrmions, and more complex structures comprised of torons, hopfions, and various disclination loops that are experimentally observed in a confinement-frustrated chiral nematic system.

Spindle-shaped nematic droplets (tactoids) form in solutions of rod-like molecules at the onset of the liquid crystalline phase. Their unique shape and internal structure result from the interplay of the elastic deformation of the nematic and anisotropic surface forces. The balance of these forces dictates that tactoids must display a continuous variation in aspect ratio and director-field configuration. Yet, such continuous transition has eluded observation for decades: tactoids have displayed either a bipolar configuration with particles aligned parallel to the droplet interface or a homogeneous configuration with particles aligned parallel to the long axis of the tactoid. Here, we report the first observation of the continuous transition in shape and director-field configuration of tactoids in true solutions of carbon nanotubes in chlorosulfonic acid. This observation is possible because the exceptional length of carbon nanotubes shifts the transition to a size range that can be visualized by optical microscopy. Polarization micrographs yield the interfacial and elastic properties of the system. Absorbance anisotropy measurements provide the highest nematic order parameter (S = 0.79) measured to date for a nematic phase of carbon nanotubes at coexistence with its isotropic phase.