#### Volume 18, issue 4 (2014)

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Ropelength criticality

### Jason Cantarella, Joseph H G Fu, Robert B Kusner and John M Sullivan

Geometry & Topology 18 (2014) 2595–2665
##### Abstract

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 bound on curvature and a self-contact condition.

We give a set of necessary and sufficient conditions for criticality with respect to this constraint, based on a version of the Kuhn–Tucker theorem that we established in previous work. 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 ${C}^{1}$–compact family of smooth functions in order to apply a theorem of Clarke. We give a number of applications, including a classification of the “supercoiled helices” formed by critical curves with no self-contacts (constrained by curvature alone) and an explicit but surprisingly complicated description of the “clasp” junctions formed when one rope is pulled tight over another.

##### Keywords
ropelength, ideal knot, tight knot, constrained minimization, Kuhn–Tucker theorem, simple clasp, Clarke gradient
##### Mathematical Subject Classification 2010
Primary: 57M25, 49J52, 53A04
##### Publication
Received: 29 November 2011
Accepted: 7 August 2012
Published: 8 December 2014
Proposed: Tobias H Colding
Seconded: Yasha Eliashberg, Dmitri Burago
##### Authors
 Jason Cantarella Department of Mathematics University of Georgia Athens, GA 30602 USA http://www.jasoncantarella.com Joseph H G Fu Department of Mathematics University of Georgia Athens, GA 30602 USA http://www.math.uga.edu/~fu/ Robert B Kusner Department of Mathematics University of Massachusetts Lederle Graduate Research Tower Box 34515 Amherst, MA 01003 USA http://www.gang.umass.edu/~kusner/ John M Sullivan Institut für Mathematik Technische Universität Berlin Str. des 17. Juni 136 10623 Berlin Germany http://www.isama.org/jms/