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...

Abstract. The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are C 1,1 curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp.

Abstract. We examine the space of surfaces in R 3 which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the space M k of all such surfaces with k ends (where surfaces are identified if they differ by an isometry of R 3 ) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has no L 2 −nullspace we prove that M k is locally the quotient of a real analytic manifold of dimension 3k − 6 by a finite group (i.e. a real analytic orbifold), for k ≥ 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension of M k is independent of the topology of the underlying punctured Riemann surface to which Σ is conformally equivalent. These results also apply to hypersurfaces of H n+1 with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.

We minimize a discrete version of the squared mean curvature integral for polyhedral surfaces in three-space, using Brakke's Surface Evolver [Brakke 1992]. Our experimental results support the conjecture that the smooth minimizers exist for each genus and are stereographic projections of certain minimal surfaces in the three-sphere.

In this paper, we study the average crossing number of equilateral random walks and polygons. We show that the mean average crossing number ACN of all equilateral random walks of length n is of the form 3 16 •n•ln n+O(n). A similar result holds for equilateral random polygons. These results are confirmed by our numerical studies. Furthermore, our numerical studies indicate that when random polygons of length n are divided into individual knot types, the ACN(K) for each knot type K can be described by a function of the form ACN(K) = a • (n − n 0) • ln(n − n 0) + b • (n − n 0) + c where a, b and c are constants depending on K and n 0 is the minimal number of segments required to form K. The ACN(K) profiles diverge from each other, with more complex knots showing higher ACN(K) than less complex knots. Moreover, the ACN(K) profiles intersect with the ACN profile of all closed walks. These points of intersection define the equilibrium length of K, i.e., the chain length ne(K) at which a statistical ensemble of configurations with given knot type K-upon cutting, equilibration and reclosure to a new knot type K-does not show a tendency to increase or decrease ACN(K). This concept of equilibrium length seems to be universal, and applies also to other length-dependent observables for random knots, such as the mean radius of gyration Rg .