arXiv research

We describe several configurations of clasped ropes which are balanced and thus critical for the Gehring ropelength problem of arXiv:math.DG/0402212.

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 Gehring's problem to deal with links in a fixed link-homotopy …

New examples show limits of physical link isotopies.

Study on Gehring link problem and width of bands in curved manifolds.

The Gehring-Martin-Tan inequality for 2-generator subgroups of PSL(2,C) is one of the best known discreteness conditions. A Kleinian group $G$ is called a Gehring-Martin-Tan group if the equality holds for the group $G$. We give a method for constructing Gehring-Martin-Tan groups with a generator of order four and pres…

In a very influential paper Gehring and Palka introduced the notions of quasiconformally homogeneous and uniformly quasiconformally homogeneous subsets of Euclidean space. Their motivation was to provide a characterization of quasi-disks, i.e. domains which are quasiconformally homeomorphic to the unit disk. As a gener…

The ropelength of a knot is the quotient of its length and 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 ropelen…

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength critical configurations for knots and links which are dif…

Investigates ropelength of complex knots and links.

Proves ropelength conjecture for alternating knots.

The ropelength of a knot is the quotient of its length by its thickness. We consider a family of energy functions for knots, depending on a power p, which approach ropelength as p increases. We describe a numerically computed trefoil knot which seems to be a local minimum for ropelength; there are nearby critical point…

The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.

New bounds on ropelength for special alternating knots.

Study quantizes ropelength and writhe of 12-crossing knots.

Using the existence of a special quadrisecant line, we show the ropelength of any nontrivial knot is at least 15.66. This improves the previously known lower bound of 12. Numerical experiments have found a trefoil with ropelength less than 16.372, so our new bounds are quite sharp.

The paper proves $\Gamma$-convergence of discrete tangent-point energies to continuous energies and ropelength, with applications to biarc curves.

The ropelength of a space curve is usually defined as the quotient of its length by its thickness: the radius of the largest embedded tube around the knot. This idea was extended to space polygons by Eric Rawdon, who gave a definition of ropelength in terms of doubly-critical self-distances (local minima of the distanc…

Optimized concentric helices minimize the ropelength of non-alternating torus knots.

For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the crossing number of K. In this paper, we show that there exists a constant a>0 such that L(K) is bounded above by a Cr(K) ln^5 (Cr(K)) for any knot K. This result shows that the upper bound of the ropelength of any knot is almost linear in terms of i…

New pseudometrics defined on knot spaces based on curve thickness and length.

New bounds on ropelength for torus links improve previous estimates.

Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of $n$-component links in terms of the Milnor linking numbers. The main goal of the…

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 kno…

For an un-oriented link $\mathcal{K}$, let $L(\mathcal{K})$ be the ropelength of $\mathcal{K}$. It is known that when $\mathcal{K}$ has more than one component, different orientations of the components of $\mathcal{K}$ may result in different braid index. We define the largest braid index among all braid indices corres…

The paper analyzes geometric densities and compression radii for knot types.

Study of Milnor invariants and ropelength of spherical links.

The paper studies knot densities under various constraints and degenerations.

Let $\mbox{Len}(K)$ be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for $\mbox{Len}(K)$ of a nontrivial knot $K$ in terms of its crossing number $c(K)$ as follows: $\mbox{Len}(K) \leq \min \left\{ \fr…

Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length $\mbox{Len}(K)$ of a knot $K$ i…

Study on knot types using thickness and length constraints.

The paper provides bounds for the ropelength of a link in terms of the crossing numbers of its split components. As in earlier papers, the bounds grow with the square of the crossing number; however, the constant involved is a substantial improvement on previous results. The proof depends essentially on writing links i…

We report on new numerical computations of the set of self-contacts in tightly knotted tubes of uniform circular cross-section. Such contact sets have been obtained before for the trefoil and figure eight knots by simulated annealing -- we use constrained gradient-descent to provide new self-contact sets for those and …

We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are defined on equilateral polygons with $n$ vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the $Γ$-limit of the di…

Optimizes the conformal capacity of linked curves in $S^3$.

There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…

The ropelength problem asks for the minimum-length configuration of a knotted diameter-one tube embedded in Euclidean three-space. The core curve of such a tube is called a tight knot, and its length is a knot invariant measuring complexity. In terms of the core curve, the thickness constraint has two parts: an upper b…

A Riemann surface $M$ is said to be $K$-quasiconformally homogeneous if for every two points $p,q \in M$, there exists a $K$-quasiconformal homeomorphism $f \colon M \rightarrow M$ such that $f(p) = q$. In this paper, we show there exists a universal constant $K_0 > 1$ such that if $M$ is a $K$-quasiconformally homogen…

The thickness, NIR(K) of a knot or link K is defined to be the radius of the largest solid tube one can put around the curve without any self intersections, which is also known as the normal injectivity radius of K. For C^{1,1} curves K, NIR(K)=min{(1/2)DCSC(K),(1/(supkappa(K))))}, where kappa(K) is the generalized cur…

We study optimal double helices with straight axes (or the fattest tubes around them) computationally using three kinds of functionals; ideal ones using ropelength, best volume packing ones, and energy minimizers using two one-parameter families of interaction energies between two strands of types $r^{-α}$ and $\frac1r…

The Moebius energy of a knot is an energy functional for smooth curves based on an idea of self-repelling. If a knot has a thick tubular neighborhood, we would intuitively expect the energy to be low. In this paper, we give explicit bounds for energy in terms of the ropelength of the knot, i.e. the ratio of the length …

A quadrisecant line is one which intersects a curve in at least four points, while an essential secant captures something about the knottedness of a knot. This survey article gives a brief history of these ideas, and shows how they may be applied to questions about the geometry of a knot via the total curvature, ropele…

It is known that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and knot we can distinguish three types of quadrisecants; the alternating ones have the most relevance for the geometry of a knot. In this paper we pro…

The paper defines and studies discrete p-density and compression-radius profiles of lattice knots.

Researchers find optimal configurations of complex knots and links.

We present new computations of tight shapes obtained using the constrained gradient descent code RIDGERUNNER for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of…

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and th…

For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r<R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds o…

In this article we survey and describe various aspects of the geometry and arithmetic of Kleinian groups - discrete nonelementary groups of isometries of hyperbolic $3$-space. In particular we make a detailed study of two-generator groups and discuss the classification of the arithmetic generalised triangle groups (and…