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Criticality for the
Gehring link problem
Jason Cantarella, Joseph H G Fu, Rob Kusner, John M Sullivan and Nancy C Wrinkle |
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Geometry & Topology 10 (2006) 2055–2115
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arXiv: math.DG/0402212 |
Abstract |
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In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring’s problem to deal with links in a fixed link-homotopy class: we prove ropelength minimizers exist and introduce a theory of ropelength criticality. Our balance criterion is a set of necessary and sufficient conditions for criticality, based on a strengthened, infinite-dimensional version of the Kuhn–Tucker theorem. We use this to prove that every critical link is with finite total curvature. The balance criterion also allows us to explicitly describe critical configurations (and presumed minimizers) for many links including the Borromean rings. We also exhibit a surprising critical configuration for two clasped ropes: near their tips the curvature is unbounded and a small gap appears between the two components. These examples reveal the depth and richness hidden in Gehring’s problem and our natural extension. |
Keywords
Gehring link problem, link homotopy, link group,
ropelength, ideal knot, tight knot, constrained
minimization, Mangasarian–Fromovitz constraint
qualification, Kuhn–Tucker theorem, simple clasp, Clarke
gradient, rigidity theory
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Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 49Q10, 53A04
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References |
Forward citations |
Publication
Received: 16 May 2005
Accepted: 23 May 2006
Published: 14 November 2006
Proposed: Yasha Eliashberg
Seconded: Joan Birman, Tobias Colding
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Authors |
| Jason Cantarella | |
| Department of Mathematics University of Georgia Athens, GA 30602 USA |
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| Joseph H G Fu | |
| Department of Mathematics University of Georgia Athens, GA 30602 USA |
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| Rob Kusner | |
| Department of Mathematics University of Massachusetts Amherst, MA 01003 USA |
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| John M Sullivan | |
| Institut für Mathematik Technische Universität Berlin DE–10623 Berlin Germany |
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| Nancy C Wrinkle | |
| Department of Mathematics Northeastern Illinois University Chicago, IL 60625 USA |
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