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Ropelength
criticality
Jason Cantarella, Joseph H G Fu, Robert B Kusner and John M Sullivan |
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Geometry & Topology 18 (2014) 2595–2665
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Abstract |
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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 –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
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Mathematical Subject Classification 2010
Primary: 57M25, 49J52, 53A04
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References |
Publication
Received: 29 November 2011
Accepted: 7 August 2012
Published: 8 December 2014
Proposed: Tobias H Colding
Seconded: Yasha Eliashberg, Dmitri Burago
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Authors |
| Jason Cantarella | |
| Department of Mathematics University of Georgia Athens, GA 30602 USA |
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| http://www.jasoncantarella.com | |
| Joseph H G Fu | |
| Department of Mathematics University of Georgia Athens, GA 30602 USA |
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| 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 |
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| http://www.gang.umass.edu/~kusner/ | |
| John M Sullivan | |
| Institut für Mathematik Technische Universität Berlin Str. des 17. Juni 136 10623 Berlin Germany |
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| http://www.isama.org/jms/ | |
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