Robert B. Kusner

On the minimum ropelength of knots and links

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 C1,1 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.

CONSTANT MEAN-CURVATURE SURFACES IN HYPERBOLIC SPACE

Supported by the National Science Foundation grant DMS-8808002. Supported by the National Science Foundation grant DMS-8908064. The research described in this paper was supported by research grant DE-FG0286ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS-8900285. Supported by the National Science Foundation grant DMS-8800414.

The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ℝ3 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 spaceMk of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ℝ3) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL2-nullspace, we prove thatMk is locally the quotient of a real analytic manifold of dimension 3k−6 by a finite group (i.e. a real analytic orbifold), fork ≥ 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 ofMk is independent of the genus of the underlying punctured Riemann surface to which Σ is conformally equivalent. These results also apply to hypersurfaces of Hn+1 with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.

Tight knot values deviate from linear relations

Applications of knots to the study of polymers have emphasized geometric measures on curves such as ‘energy’ and ‘rope length’, which, when minimized over different configurations of a knot, give computable knot invariants related to physical quantities. In DNA knots, electrophoretic mobility appears to be correlated with the average crossing number of rope-length-minimizing configurations, and a roughly linear empirical relation has been observed between the crossing number and rope length. Here we show that a linear relation cannot hold in general, and we construct infinite families of knots whose rope length grows as the 3/4 power of the crossing number. It can be shown that no smaller power is possible.

Conformal geometry and complete minimal surfaces

1. Introduction. This note applies new ideas from conformai geometry to the study of complete minimal surfaces of finite total curvature in R 3. The two main results illustrate this in complementary ways. Theorem A implies several uniqueness or nonexistence corollaries, while Theorem B constructs new examples, including the first complete immersed nonorientable minimal surfaces with finite total curvature and embedded ends. This construction is based upon the close relationship between these minimal surfaces and certain compact immersed surfaces minimizing the conformally invariant functional W = fH 2 da (Theorems C and D).

Criticality for the Gehring link problem

In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehrings problem to deal with links in a fixed link-homotopy class: we prove ropelength minimizers exist and introduce a theory of ropelength criticality. Our balance criterion is a set of necessary and sufficient conditions for criticality, based on a strengthened, infinite-dimensional version of the Kuhn--Tucker theorem. We use this to prove that every critical link is C^1 with finite total curvature. The balance criterion also allows us to explicitly describe critical configurations (and presumed minimizers) for many links including the Borromean rings. We also exhibit a surprising critical configuration for two clasped ropes: near their tips the curvature is unbounded and a small gap appears between the two components. These examples reveal the depth and richness hidden in Gehrings problem and our natural extension.

Triunduloids: embedded constant mean curvature surfaces with three ends and genus zero

We construct the entire three-parameter family of embedded constant mean curvature surfaces with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the necksizes of the three ends. Because our surfaces are transcendental, and are not described by any ordinary dierential equation, it is remarkable to obtain such an explicit determination of their moduli space. (AMS Classification 2000: 53A10, 58D10)

The average crossing number of equilateral random polygons

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 . 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 for each knot type can be described by a function of the form where a, b and c are constants depending on and n0 is the minimal number of segments required to form . The profiles diverge from each other, with more complex knots showing higher than less complex knots. Moreover, the profiles intersect with the ACN profile of all closed walks. These points of intersection define the equilibrium length of , i.e., the chain length at which a statistical ensemble of configurations with given knot type —upon cutting, equilibration and reclosure to a new knot type —does not show a tendency to increase or decrease . 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.

The minimax sphere eversion

We consider an eversion of a sphere driven by a gradient flow for elastic bending energy. We start with a halfway model which is an unstable Willmore sphere with 4-fold orientation-reversing rotational symmetry. The regular homotopy is automatically generated by flowing down the gradient of the energy from the halfway model to a round sphere, using the Surface Evolver. This flow is not yet fully understood; however, our numerical simulations give evidence that the resulting eversion is isotopic to one of Morin’s classical sphere eversions. These simulations were presented as real-time interactive animations in the CAVE TM automatic virtual environment at Supercomputing’95, as part of an experiment in distributed, parallel computing and broad-band, asynchronous networking.

The number of faces in a minimal foam

We derive a formula for the average number of faces per cell in a foam of minimal surfaces. In case of periodic minimal foam in R3 this formula implies that the fundamental cell has at least 14 faces, a lower bound achieved by Kelvin’s example from the last century.