Ropelength Criticality

TL;DR

This paper characterizes the conditions for a knotted tube configuration to be length-minimizing under thickness constraints, providing new theoretical insights and classifications of critical configurations like supercoiled helices and clasp junctions.

Contribution

It establishes necessary and sufficient conditions for criticality in the ropelength problem using advanced variational techniques and Clarke's theorem, advancing understanding of tight knot configurations.

Findings

Classification of supercoiled helices without self-contacts

Explicit description of clasp junctions in tight configurations

Development of a derivative formula for thickness under perturbations

Abstract

The ropelength problem asks for the minimum-length configuration of a knotted diameter-one tube embedded in Euclidean three-space. 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. We give a set of necessary and sufficient conditions for criticality with respect to this constraint, based on a version of the Kuhn-Tucker theorem that we established in previous work. 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. We give a number of applications, including 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 the "clasp" junctions formed when one rope is pulled tight over another.