Total Curvature and Packing of Knots

We establish a new fundamental relationship between total curvature of knots and crossing number. If K is a smooth knot in R3, R the cross-section radius of a uniform tube neighborhood K, L the arclength of K, and κ the total curvature of K, then (up to some coefficient), crossing number of K ≤ L R κ . The proof generalizes to show that for smooth knots in R3, the crossing number, writhe, Möbius Energy, Normal Energy, and Symmetric Energy each is bounded by the product of total curvature and rope-length. One can construct knots in which the crossing numbers grow as fast as the (4/3) power of L R . Our theorem says that such families must have unbounded total curvature: If the total curvature is bounded, then the rate of growth of crossings with ropelength can only be linear. On the way to this theorem, we establish fundamental lemmas about the total curvature of curves that are packed in certain ways: If a long smooth curve A with arclength L is contained in a solid ball of radius ρ, then the total curvature of K is at least proportional to L/ρ. If A connects concentric spheres of radii a ≥ 2 and b ≥ a + 1, by running from the inner sphere to the outer sphere and back again, then the total curvature of A is at least proportional to 1/ √ a. Department of Mathematics, St. Anselm College, Manchester, NH. Research supported by NSF Grant #DMS0107747 Department of Mathematics, University of Iowa, Iowa City IA 52240. Research supported by NSF Grant #DMS0107209. We thank J. McAtee and R. Weiler for helpful comments