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Ropelength Criticality

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

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.Comment: 72 pages, 10 figures; v3: incorporate referee's comments: minor fixes; expository improvements; slight strengthening of some result

Topics: Mathematics - Differential Geometry, Mathematics - Geometric Topology, 53A04, 57M25, 49J52
Year: 2013
DOI identifier: 10.2140/gt.2014.18.1973
OAI identifier: oai:arXiv.org:1102.3234
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