The Study of Ropelength and Knot Energies

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

DMS-0204826 Jason Cantarella and Joseph Fu The project will study the geometry of knots which are critical configurations for "ropelength", which is defined to be the quotient of their length over the radius of their largest embedded tubular neighborhood. Since the ropelength functional is not smooth, there is no classical Euler-Lagrange equation describing the minimizers. The primary aim of the projec666t is to provide an analogy to the Euler-Lagrange equation: a "balance criterion", which shows that a curve is ropelength critical if and only if the first variation of the length of the core curve can be balanced by a system of self-contact forces acting on the surrounding tube. The methods used to prove the balance criterion (a generalization of certain aspects of tensegrity theory to a class of "continuous tensegrities", using a version of the Farkas alternative theorem for linear operators on Banach spaces) promise rich applications in other areas of mathematics, such as the study of convex curves, and of "knot energies". A natural model for a rope with a circular cross-section is a space curve surrounded by a non-self-intersecting tube of fixed radius. If such a rope is tied in a knot, and the knot is pulled tight, the resulting curve is a critical configuration for the "ropelength" of the curve, which is defined to be the quotient of the length of the curve over the radius of the tube. The project studies the geometry of ropelength-critical curves using ideas from the theory of frameworks. This theory studies simple engineering models of structures, and gives a precise description of how external loads on a structure are resolved into tensions and compressions of different structural elements. The project views the tension in a tight knot as exerting a force directed towards the inside of each curve of the rope, and aims to show that such a curve is tight if this force can be resolved into a system of self-contact forces acting on the surface of the tube. This "balance criterion" has surprising consequences. Imagine a rope stretched horizontally, like a clothesline. If another rope is passed over the line, and pulled down until the pair is tight, the ropes form four straight segments joined to a central region where the strands curve around one another. One would expect the ropes to maintain contact throughout this curved clasp, with the two points on the inside of each bend touching one another. However, according to the balance criterion, this cannot happen: there is always a small gap between the two strands at the center of the turn. This model has already been used in molecular biology to describe the behavior of DNA strands; the new geometric information provided by the project will help to refine and extend many other appications of this model in physics, biology, and engineering.