Characterization of ideal knots |
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Authors: | Email author" target="_blank">Friedemann?SchurichtEmail author Heiko?von der Mosel |
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Institution: | (1) Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany;(2) Mathematisches Institut, Universität Bonn, Beringstraße 1, 53115 Bonn, Germany |
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Abstract: | We present a characterization of ideal knots, i.e., of closed knotted curves of prescribed thickness with minimal length, where we use the notion of global curvature for the definition of thickness. We show with variational methods that for an ideal knot
, the normal vector
at a curve point
is given by the integral over all vectors
against a Radon measure, where
realizes the given thickness. As geometric consequences we obtain in particular, that points without contact lie on straight segments of
, and for points
with exactly one contact point
we have that
points exactly into the direction of
Moreover, isolated contact points lie on straight segments of
, and curved arcs of
consist of contact points only, all realizing the prescribed thickness with constant (maximal) global curvature.Received: 1 January 2003, Accepted: 12 March 2003, Published online: 1 July 2003Mathematics Subject Classification (2000):
53A04, 57M25, 74K05, 74M15, 92C40 |
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Keywords: | |
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