Polygonal Approximation and Energy of Smooth Knots (original) (raw)
Related papers
POLYGONAL KNOT SPACE NEAR ROPELENGTH-MINIMIZED KNOTS
Journal of Knot Theory and Its Ramifications, 2008
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 on perturbation radii for which we can see small trefoil and figure-eight summands and use Monte Carlo simulations to approximate the relative probabilities of these structures as a function of the number of edges.
An Investigation of Polygonal Knot Spaces and Ideal Physical Knot Configurations
Spaces of polygonal knots, subject to specified constraints such as the number of nondegenerate edges or the requirement of having fixed edge lengths, provide the context within which it is appropriate to study configurations which are ideal with respect to a variety of natural physically motivated constraints. Even for polygonal knots with relatively few vertices, the high dimensionality and complexity of the knot space structure makes analytical investigations impractical. In this note we will discuss the methods and the results of a Monte Carlo investigation of several fundamental approaches to ideal polygonal knot configurations for small numbers of edges.
Monte Carlo explorations of polygonal knot spaces
Knots in Hellas, 2000
Polygonal knots are embeddings of polygons in three space. For each n, the collection of embedded n-gons determines a subset of Euclidean space whose structure is the subject of this paper. Which knots can be constructed with a specified number of edges? What is the likelihood that a randomly chosen polygon of n-edges will be a knot of a specific topological type? At what point is a given topological type most likely as a function of the number of edges? Are the various orderings of knot types by means of "physical properties" comparable? These and related questions are discussed and supporting evidence, in many cases derived from Monte Carlo explorations, is provided.
Topology and its Applications, 2002
The Möbius 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 of a thickest tube to its radius.
Ropelength of tight polygonal knots
2005
A physical interpretation of the rope simulated by the SONO algorithm is presented. Properties of the tight polygonal knots delivered by the algorithm are analyzed. An algorithm for bounding the ropelength of a smooth inscribed knot is shown. Two ways of calculating the ropelength of tight polygonal knots are compared. An analytical calculation performed for a model knot shows that an appropriately weighted average should provide a good estimation of the minimum ropelength for relatively small numbers of edges.
Russian Journal of Mathematical Physics, 2011
We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals considered are the sum of two terms, the uniformization term (which tends to make the curvature of the knot uniform) and the resistance term (which, in particular, forbids crossing changes). We define an infinite family of uniformization functionals, depending on an arbitrary smooth function f and study the simplest nontrivial case f (x) = x 2 , obtaining neat normal forms (corresponding to minima of the functional) by making use of the Gauss representation of immersed curves, of the phase space of the pendulum, and of elliptic functions.
Knotted polygons with curvature in
Journal of Physics A: Mathematical and General, 1998
The knot probability of semiflexible polygons on the cubic lattice is investigated. The degree of stiffness of the polygon is mimicked by introducing a bending fugacity conjugate to the curvature of the polygon. By generalizing Kesten's pattern theorem to semiflexible walks, we show that for any finite value of the bending fugacity all except exponentially few sufficiently long polygons are knotted.
Knots, Slipknots, and Ephemeral Knots in Random Walks and Equilateral Polygons
2008
The probability that a random walk or polygon in the 3-space or in the simple cubic lattice contains a knot goes to one at the length goes to infinity. Here, we prove that this is also true for slipknots consisting of unknotted portions, called the slipknot, that contain a smaller knotted portion, called the ephemeral knot. As is the case with knots, we prove that any topological knot type occurs as the ephemeral knotted portion of a slipknot. 1
Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ³
Contemporary Mathematics, 2002
Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society,
Total curvature and packing of knots
Topology and its Applications, 2007
We establish a new relationship between total curvature of knots and crossing number. If K is a smooth knot in R 3 , R the cross-section radius of a uniform tube neighborhood K, L the arclength of K, and κ the total curvature of K, then crossing number of K < 4 L R κ. The proof generalizes to show that for smooth knots in R 3 , the crossing number, writhe, Möbius Energy, Normal Energy, and Symmetric Energy are all bounded by the product of total curvature and rope-length. One can construct knots in which the crossing numbers grow as fast as the (4/3) power of L R. Our theorem says that such families must have unbounded total curvature: If the total curvature is bounded, then the rate of growth of crossings with ropelength can only be linear. Our proof relies on fundamental lemmas about the total curvature of curves that are packed in certain ways: If a long smooth curve A with arclength L is contained in a solid ball of radius ρ, then the total curvature of K is at least proportional to L/ρ. If A connects concentric spheres of radii a ≥ 2 and b ≥ a + 1, by running from the inner sphere to the outer sphere and back again, then the total curvature of A is at least proportional to 1/ √ a.