Criticality for the Gehring link problem
By Jason H. Cantarella, Joseph H. G. Fu, Robert B. Kusner, John M. Sullivan Jr., Nancy C. Wrinkle. 2004
68 citations recorded by the source corpus.
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 C^1 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.
Subfields: Civil and Structural Engineering, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/460f784272e1e874e840807acd1e09ae47d42fb8
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In Computational Theory and Mathematics
In Geometry and Topology
Citations in the corpus
- Regularizing and self-avoidance effects of integral Menger curvature (2010, unknown): “However, the thickness [·] depends not only highly nonlinearly but also in a nonsmooth fashion on the curve γ , a fact which turned out to be a challenge for regularity investigations [30, 31, 4], and for numerical treatment [1, 6, 7].”
- On rectifiable curves with Lp-bounds on global curvature: self-avoidance, regularity, and minimizing knots (2007, background): “4), turned out to be quite challenging, see [42], [43], [7], [14].”
- What are the Longest Ropes on the Unit Sphere? (2011, unknown): “see [2,7,21].”