#### Volume 10, issue 4 (2006)

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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

Geometry & Topology 10 (2006) 2055–2115
 arXiv: math.DG/0402212
##### Abstract

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.

##### 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
##### Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 49Q10, 53A04