Ropelength criticality
type: Item
instance of: scholarly article
title: Ropelength criticality
zbMATH Open document ID: 1305.57008
author: Jason Cantarella
author: Robert B. Kusner
author: John M. Sullivan
author: Joseph H. G. Fu
published in: Geometry \& Topology
publication date: 2014-10-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.3234
reviewed by: Claus Ernst
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 6356604
zbMATH Keywords: ropelength
zbMATH Keywords: ideal knot
zbMATH Keywords: tight knot
zbMATH Keywords: constrained minimization
zbMATH Keywords: reach
zbMATH Keywords: Kuhn-Tucker theorem
zbMATH Keywords: simple clasp
zbMATH Keywords: Clarke gradient
MaRDI profile type: Publication
OpenAlex ID: W3037556638
arXiv ID: 1102.3234
cites work: Q5477844
cites work: Q4115334
cites work: Symmetric criticality for tight knots
cites work: Criticality for the Gehring link problem
cites work: On the minimum ropelength of knots and links
cites work: Q4733008
cites work: Generalized Gradients and Applications
cites work: Topological and physical link theory are distinct
cites work: Quadrisecants give new lower bounds for the ropelength of a knot
cites work: Distributions
cites work: LOCAL STRUCTURE OF IDEAL KNOTS, II CONSTANT CURVATURE CASE
cites work: Curvature Measures
cites work: Interior Methods for Nonlinear Optimization
cites work: Global curvature, thickness, and the ideal shapes of knots
cites work: Q5566063
cites work: Ropes in Equilibrium
cites work: High resolution portrait of the ideal trefoil knot
cites work: Q3996430
cites work: Characterization of ideal knots
cites work: A constructive approach to modelling the tight shapes of some linked structures
cites work: Regularizing and self-avoidance effects of integral Menger curvature
cites work: Q3499992
DOI: 10.2140/GT.2014.18.1973
review text: The goal of this paper is to investigate what shape a knot or link attains when it is tied in rope of a given diameter (or thickness) and then pulled tight. This is called the \textit{ropelength problem}: to minimize the length of a knot or link \(L\) in Euclidean space subject to the condition that it remains one unit thick. 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 bound on curvature and a self-contact condition. This article extends earlier work by the four authors [Geom. Topol. 10, 2055--2115 (2006; Zbl 1129.57006)], where they studied a simplified version, the Gehring link problem, in which the thickness constraint is replaced by the weaker requirement that the link-thickness - the minimal distance between different components of the link - is at least 1. In this reference the authors made precise the intuition that for the Gehring problem, in a critical configuration for a link \(L\), the tension forces seeking to minimize length must be balanced by contact forces. More precisely, a strut is a pair of points on different components at distance exactly 1. The balance criterion says that \(L\) is critical if and only if there is a nonnegative measure on the set of struts, thought of as a system of compression forces, which balances the curvature vector field of \(L\).NEWLINENEWLINEIn the present paper the authors adopt the same general approach to develop a criticality theory for the (technically much more difficult) ropelength problem. The authors give a set of necessary and sufficient conditions for criticality with respect to the thickness constraint, based on a version of the Kuhn-Tucker theorem that was established previously by the authors. The key technical difficulty is to compute the derivative of thickness under a smooth perturbation. This is accomplished by writing thickness as the minimum of a \(C^1\)-compa…
description: scientific article; zbMATH DE number 6356604