Total Curvature and Packing of Knots
Geometric Topology
2007-05-23 v2 Differential Geometry
Abstract
We establish a new fundamental relationship between total curvature of knots and crossing number. If K is a smooth knot in 3-space, R the cross-section radius of a uniform tube neighborhood of K, L the arclength of K, and k the total curvature of K, then (up to a coefficient independent of K), crossing number of K < (k)(L/R). There are families of knots whose crossing numbers grow faster than either k or L/R separately. For example, the knots whose crossing numbers grow with the (4/3)-power of ropelength must have total curvature growing arbitrarily large as well.
Cite
@article{arxiv.math/0310365,
title = {Total Curvature and Packing of Knots},
author = {Gregory Buck and Jonathan Simon},
journal= {arXiv preprint arXiv:math/0310365},
year = {2007}
}
Comments
19 pages, no figures. This update of the Oct. 03 version has improved Lemma 1.1 and resulting improved coefficients