Local Structure of Ideal Shapes of Knots

Details Local Structure of Ideal Shapes of Knots Local Structure of Ideal Shapes of Knots

2002-04-04

arxiv.org/abs/math/0204063v1

Mathematics

Geometric Topology

Scientific paper

Relatively extremal knots are the relative minima of the ropelength functional in C^1 topology. On the set curves of fixed length, they are the relative maxima of thickness (normal injectivity radius) functional, including the ideal knots. We prove that a C^{1,1} relatively extremal knot in R^n has thickness equal to half of the minimal double critical distance unless it has constant maximal (generalized) curvature. This result also applies to links since our method is local. Our main approach is to show that the shortest curves with bounded curvature and C^1 boundary conditions in R^n contain CLC (circle-line-circle) curves, if they do not have constant maximal curvature.

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