Total Curvature and Packing of Knots

Details Total Curvature and Packing of Knots Total Curvature and Packing of Knots

2003-10-22

arxiv.org/abs/math/0310365v2

Mathematics

Geometric Topology

19 pages, no figures. This update of the Oct. 03 version has improved Lemma 1.1 and resulting improved coefficients

Scientific paper

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.

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