Thickness of knots
classical knot theory, knots, ropelength
Classical knot theory studies one-dimensional filaments; in this paper we model knots as more physically “real”, e.g., made of some “rope” with nonzero thickness. A motivating question is: How much length of unit radius rope is needed to tie a nontrivial knot? For a smooth knot K, the “injectivity radius” R( K) is the supremum of radii of embedded tubular neighborhoods. The “thickness” of K, a new measure of knot complexity, is the ratio of R( K) to arc-length. We relate thickness to curvature, self-distance, distortion, and (for knot types) edge-number.
Topology and its Applications