201. Thickness of knots
- Author
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Oguz C. Durumeric, Eric J. Rawdon, Jonathan Simon, and R. A. Litherland
- Subjects
Ropelength ,Knots ,Curvature ,Fibered knot ,Geometry ,Distortion ,Self-distance ,Edge-number ,Knot energy ,Mathematics::Geometric Topology ,Knot theory ,Knot (unit) ,Geometry and Topology ,Thickness ,Trefoil knot ,Rope ,Mathematics - Abstract
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.
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