相关论文: Local Structure of Ideal Shapes of Knots
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,…
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…
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…
What length of rope (of given diameter) is required to tie a particular knot? To answer this question, we define some new notions of thickness for a space curve, one based on Gromov's distortion, and another generalizing the thickness of…
We study simple, knotted and linked torus windings that are made of tubes of finite thickness. Knots which have the shortest rope length are often denoted ideal structures. Conventionally, the ideal structure are found by rope shortening…
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…
We perform a compare-and-contrast investigation between the equilibrium shapes of physical and ideal trefoil knots, both in closed and open configurations. Ideal knots are purely geometric abstractions for the tightest configuration tied in…
In this note, we attempt to find counterexamples to the conjecture that the ideal form of a knot, that which minimizes its contour length while respecting a no-overlap constraint, also minimizes the volume of the knot, as determined by its…
The problem of finding robust and effective methods for locating entanglement in embedded curves is relevant to both applications and theoretical investigations. Rather than focusing on an exact determination, we introduce the knot…
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…
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…
We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We…
Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise…
We show that smooth curves with prescribed curvature satisfy a $C^1$-dense $h$-principle in the space of immersed curves in Euclidean space. More precisely, every $C^{\alpha \geq 2}$ curve with nonvanishing curvature in $R^{n\geq 3}$ can be…
The ropelength of a knot is the minimum contour length of a tube of unit radius that traces out the knot in three dimensional space without self-overlap, colloquially the minimum amount of rope needed to tie a given knot. Theoretical upper…
The $c$-curvature of a complete surface with Gauss curvature close to 1 in $C^2$ norm is almost-positive (in the sense of Kim--McCann). Our proof goes by a careful case by case analysis combined with perturbation arguments from the constant…
In this paper we show how to realize all knot (and link) types as C^{2} smooth curves of constant curvature. Our proof is constructive: we build the knots with copies of a fixed finite number of "building blocks" that are particular…
It is well-known that normal extremals in sub-Riemannian geometry are curves which locally minimize the energy functional. Most proofs of this fact do not make, however, an explicit use of relations between local optimality and the geometry…
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…
Ropelength, L, is a parameter characterizing the minimum contour length of a knot or link. There exist upper and lower bounds on ropelength with respect to crossing number, C, including a universal lower bound constraining $L\geq\alpha_0…