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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

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 C1–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.

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