Ropelength criticality
The goal of this paper is to investigate what shape a knot or link attains when it is tied in rope of a given diameter (or thickness) and then pulled tight. This is called the \textit{ropelength problem}: to minimize the length of a knot or link \(L\) in Euclidean space subject to the condition that it remains one unit thick. The core curve of such a tube is called a tight knot, and its length is a knot invariant measuring complexity. In terms of the core curve, the thickness constraint has two parts: an upper bound on curvature and a self-contact condition. This article extends earlier work by the four authors [Geom. Topol. 10, 2055--2115 (2006; Zbl 1129.57006)], where they studied a simplified version, the Gehring link problem, in which the thickness constraint is replaced by the weaker requirement that the link-thickness - the minimal distance between different components of the link - is at least 1. In this reference the authors made precise the intuition that for the Gehring problem, in a critical configuration for a link \(L\), the tension forces seeking to minimize length must be balanced by contact forces. More precisely, a strut is a pair of points on different components at distance exactly 1. The balance criterion says that \(L\) is critical if and only if there is a nonnegative measure on the set of struts, thought of as a system of compression forces, which balances the curvature vector field of \(L\).NEWLINENEWLINEIn the present paper the authors adopt the same general approach to develop a criticality theory for the (technically much more difficult) ropelength problem. The authors give a set of necessary and sufficient conditions for criticality with respect to the thickness constraint, based on a version of the Kuhn-Tucker theorem that was established previously by the authors. The key technical difficulty is to compute the derivative of thickness under a smooth perturbation. This is accomplished by writing thickness as the minimum of a \(C^1\)-compact family of smooth functions in order to apply a theorem of Clarke.NEWLINENEWLINEIn the second half of the paper the authors give a number of applications: A classification of the ``supercoiled helices'' formed by critical curves with no self-contacts (constrained by curvature alone) and an explicit but surprisingly complicated description of a variety of clasps. A clasp is the simplest configuration of two (non-trivial) interlooped ropes, one anchored to the floor and one to the ceiling (i.e two parallel planes) or more generally where the four endpoints of the ropes make an angle \(\tau\) with the horizontal plane.
scientific article; zbMATH DE number 6356604
scientific article