Jason Cantarella

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.

Vector Calculus and the Topology of Domains in 3-Space

Suppose you have a vector field defined on a bounded domain in 3-space. How can you tell whether your vector field is the gradient of some function? Or the curl of another vector field? Can you find a nonzero field on your domain that is divergence-free, curlfree, and tangent to the boundary? How about a nonzero field that is divergence-free, curl-free, and orthogonal to the boundary? To answer these questions, you need to understand the relationship between the calculus of vector fields and the topology of their domains of definition. The Hodge Decomposition Theorem provides the key by decomposing the space of vector fields on the domain into five mutually orthogonal subspaces that are topologically and analytically meaningful. This decomposition is useful not only in mathematics, but also in fluid dynamics, electrodynamics, and plasma physics. Furthermore, carrying out the proof provides a pleasant introduction to homology and cohomology theory in a familiar setting, and a chance to see both the general Hodge theorem and the de Rham isomorphism theorem in action. Our goal is to give an elementary exposition of these ideas. We provide three sections of background information early in the paper: one on solutions of the Laplace and Poisson equations with Dirichlet and Neumann boundary conditions, one on the Biot-Savart law from electrodynamics, and one on the topology of compact domains in 3-space. Near the end, we see how everything we have learned provides answers to the four questions that we have posed. We close with a brief survey of the historical threads that led to the Hodge Decomposition Theorem, and a guide to the literature.

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.

The Biot–Savart operator for application to knot theory, fluid dynamics, and plasma physics

The writhing number of a curve in 3-space is the standard measure of the extent to which the curve wraps and coils around itself; it has proved its importance for molecular biologists in the study of knotted DNA and of the enzymes which affect it. The helicity of a vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid dynamics and plasma physics. The Biot–Savart operator associates with each current distribution on a given domain the restriction of its magnetic field to that domain. When the domain is simply connected, the divergence-free fields which are tangent to the boundary and which minimize energy for given helicity provide models for stable force-free magnetic fields in space and laboratory plasmas; these fields appear mathematically as the extreme eigenfields for an appropriate modification of the Biot–Savart operator. Information about these fields can be converted into bounds on the writhing number of a given piece of DNA. The purpose of this paper is to reveal new properties of the Biot–Savart operator which are useful in these applications.

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.

Knot Tightening by Constrained Gradient Descent

We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of “tight” knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their knot types. While previous authors have minimized ropelength for polygons using simulated annealing, the new idea in our code is to minimize length over the set of polygons of thickness at least one using a version of constrained gradient descent. We rewrite the problem in terms of minimizing the length of the polygon subject to an infinite family of differentiable constraint functions. We prove that the set of variations of a polygon of thickness one that does not decrease thickness to first order is a finitely generated polyhedral cone, and give an explicit set of generators. Using this cone, we give a first-order minimization procedure and a Karush–Kuhn–Tucker criterion for polygonal-ropelength criticality. Our main numerical contribution is a set of 379 almost-critical knots and links, including all prime knots with ten and fewer crossings and all prime links with nine and fewer crossings. For links, these are the first published ropelength figures, and for knots they improve on existing figures. We give new maps of the self-contacts of these knots and links, and discover some highly symmetric tight knots with particularly simple-looking self-contact maps.

The spectrum of the curl operator on spherically symmetric domains

This paper presents a mathematically complete derivation of the minimum-energy divergence-free vector fields of fixed helicity, defined on and tangent to the boundary of solid balls and spherical shells. These fields satisfy the equation ∇×V=λV, where λ is the eigenvalue of curl having smallest nonzero absolute value among such fields. It is shown that on the ball the energy minimizers are the axially symmetric spheromak fields found by Woltjer and Chandrasekhar–Kendall, and on spherical shells they are spheromak-like fields. The geometry and topology of these minimum-energy fields, as well as of some higher-energy eigenfields, are illustrated.

Isoperimetric problems for the helicity of vector fields and the Biot–Savart and curl operators

The helicity of a smooth vector field defined on a domain in three-space is the standard measure of the extent to which the field lines wrap and coil around one another. It plays important roles in fluid mechanics, magnetohydrodynamics, and plasma physics. The isoperimetric problem in this setting is to maximize helicity among all divergence-free vector fields of given energy, defined on and tangent to the boundary of all domains of given volume in three-space. The Biot–Savart operator starts with a divergence-free vector field defined on and tangent to the boundary of a domain in three-space, regards it as a distribution of electric current, and computes its magnetic field. Restricting the magnetic field to the given domain, we modify it by subtracting a gradient vector field so as to keep it divergence-free while making it tangent to the boundary of the domain. The resulting operator, when extended to the L2 completion of this family of vector fields, is compact and self-adjoint, and thus has a largest ...

Visualizing the tightening of knots

The study of physical models for knots has recently received much interest in the mathematics community. In this paper, we consider the ropelength model, which considers knots tied in an idealized rope. This model is interesting in pure mathematics, and has been applied to the study of a variety of problems in the natural sciences as well. Modeling and visualizing the tightening of knots in this idealized rope poses some interesting challenges in computer graphics. In particular, self-contact in a deformable rope model is a difficult problem which cannot be handled by standard techniques. In this paper, we describe a solution based on reformulating the contact problem and using constrained-gradient techniques from nonlinear optimization. The resulting animations reveal new properties of the tightening flow and provide new insights into the geometric structure of tight knots and links.

Probability Theory of Random Polygons from the Quaternionic Viewpoint

We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Hausmann and Knutson, using the Hopf map on quaternions from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons that comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributions. This makes our polygon measures different from those usually studied, which have Gaussian or fixed edgelengths. One advantage of our measures is that we can explicitly compute expectations and moments for chord lengths and radii of gyration. Another is that direct sampling according to our measures is fast (linear in the number of edges) and easy to code. Some of our methods will be of independent interest in studying other probability measures on polygon spaces. We define an edge set ensemble (ESE) to be the set of polygons created by rearranging a given set of n edges. A key theorem gives a formula for the average over an ESE of the squared lengths of chords skipping k vertices in terms of k, n, and the edgelengths of the ensemble. This allows one to easily compute expected values of squared chord lengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges.