Mathematics – Differential Geometry
Scientific paper
2011-02-16
Mathematics
Differential Geometry
53 pages, 9 figures; v2: minor fixes; major expository improvements; slight strengthening of some results
Scientific paper
The ropelength problem asks for the minimum-length configuration of a knotted tube embedded with fixed diameter. 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 critical curves with no self-contacts (constrained by curvature alone), a characterization of helical segments in tight links, and an explicit but surprisingly complicated description of tight clasps.
Cantarella Jason
Fu Joseph H. G.
Kusner Robert
Sullivan John M.
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