Local structure of ideal shapes of knots

Abstract

Relatively extremal knots are the relative minima of the ropelength functional in the C 1 topology. They are the relative maxima of the thickness (normal injectivity radius) functional on the set of curves of fixed length, and they include the ideal knots. We prove that a C 1 , 1 relatively extremal knot in R n either has constant maximal (generalized) curvature, or its thickness is equal to half of the double critical self distance. This local result also applies to the links. 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.