Geodesic
Ropelength criticality
Cantarella, Jason1 ; Fu, Joseph H G1 ; Kusner, Robert B2 ; Sullivan, John M3
1 Department of Mathematics, University of Georgia, Athens, GA 30602, USA
2 Department of Mathematics, University of Massachusetts, Lederle Graduate Research Tower, Box 34515, Amherst, MA 01003, USA
3 Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany
Geometry & topology, Tome 18 (2014) no. 4, pp. 2595-2665.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/gt.2014.18.1973
Classification : 57M25, 49J52, 53A04
Keywords: ropelength, ideal knot, tight knot, constrained minimization, Kuhn–Tucker theorem, simple clasp, Clarke gradient

Cantarella, Jason 1 ; Fu, Joseph H G 1 ; Kusner, Robert B 2 ; Sullivan, John M 3

1 Department of Mathematics, University of Georgia, Athens, GA 30602, USA
2 Department of Mathematics, University of Massachusetts, Lederle Graduate Research Tower, Box 34515, Amherst, MA 01003, USA
3 Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany 
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Cantarella, Jason; Fu, Joseph H G; Kusner, Robert B; Sullivan, John M. Ropelength criticality. Geometry & topology, Tome 18 (2014) no. 4, pp. 2595-2665. doi : 10.2140/gt.2014.18.1973. http://geodesic.mathdoc.fr/articles/10.2140/gt.2014.18.1973/

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