Total curvature and packing of knots

Abstract

We establish a new relationship between total curvature of knots and crossing number. If K is a smooth knot in R 3 , R the cross-section radius of a uniform tube neighborhood K, L the arclength of K, and κ the total curvature of K, then crossing number of K < 4 L R κ . The proof generalizes to show that for smooth knots in R 3 , the crossing number, writhe, Möbius Energy, Normal Energy, and Symmetric Energy are all 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. Our proof relies on 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 .