  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.
