Geometry & Topology 10 (2006) 2055–2115

DOI: 10.2140/gt.2006.10.2055

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 C¹ 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

Primary: 57M25

Secondary: 49Q10, 53A04

References

Publication

Received: 16 May 2005
Accepted: 23 May 2006
Published: 14 November 2006
Proposed: Yasha Eliashberg
Seconded: Joan Birman, Tobias Colding

Authors