Criticality for the Gehring link problem

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 (Theorem 5.4) of the KuhnTucker 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.

A Fast Octree-Based Algorithm for Computing Ropelength

“…As mentioned in the introduction, Cantarella, Fu, Kusner, Sullivan, and Wrinkle [4] have discovered tiny straight segments in a ropelength-critical simple clasp. These segments are a few one-thousands of one unit in length out of a total clasp length of about 6 units (a similar clasp has been constructed by Starostin [20]).…”

“…In the past decade, there has been a great deal of interest in exploring the geometry of tight knots; the definition of thickness has been refined and fully understood [10], it has been shown that C 1,1 minimizers exist in each knot type [5,8,9], some minimizing links have been found [5], and a theory of ropelength criticality has started to emerge [4,21]. The development of this theory has been fueled by a steady stream of numerical data on ropelength minimizers, from Pieranski's original SONO algorithm [15] and Rawdon's TOROS [16], to second-generation efforts such as Smutny and Maddocks' biarc computations [6,19] and the RIDGERUNNER project of Cantarella, Piatek, and Rawdon.…”

Some Ropelength-Critical Clasps

“…There are many ways to measure the thickness of a space curve, but for links one particularly simple notion, the Gehring thickness, is simply the minimum distance between different components. With Cantarella, Fu and Kusner we introduced a theory of criticality [2] for the Gehring ropelength problem. Our necessary and sufficient conditions for ropelength criticality take the form of a balance criterion which says that the tension force trying to reduce the length of the curves must be balanced by contact forces acting at points achieving the minimum distance.…”

“…Our necessary and sufficient conditions for ropelength criticality take the form of a balance criterion which says that the tension force trying to reduce the length of the curves must be balanced by contact forces acting at points achieving the minimum distance. One simple example, with surprising intricacy for its solution [2], is the clasp. This is a generalized link whose components are not closed curves but instead have constrained endpoints.…”

“…Here we recall the necessary definitions and theorems from our work [2] with Cantarella, Fu and Kusner. These will show the balance criterion for Gehring ropelength in the form in which we apply it to our new examples.…”