Volume 18, issue 4 (2014)

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Ropelength criticality

Jason Cantarella, Joseph H G Fu, Robert B Kusner and John M Sullivan

Geometry & Topology 18 (2014) 2595–2665
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.

Keywords
ropelength, ideal knot, tight knot, constrained minimization, Kuhn–Tucker theorem, simple clasp, Clarke gradient
Mathematical Subject Classification 2010
Primary: 57M25, 49J52, 53A04