Thickness and crossing number of knots

Given a "short" piece of rope, one can tie only "simple" knots. We make this precise by modeling "rope" as a solid tube of constant radius about a smooth care. The complexity of a knot is captured by its average crossing number which in turn bounds the minimum crossing number for the knot type. Then the ratio, L, of rope-length to radius provides an upper bound for the crossing number. Our bound is in terms of L-4/3, which we believe is the lowest exponent possible. Our route for connecting rope-length of a knot to its thickness is via self-repelling knot energies, the normal energy E-N (K) and the symmetric energy E-S(K). (C) 1999 Elsevier Science B.V. All rights reserved.