Thickness and crossing number of knots

AbstractGiven 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 core. 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 L43 , 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 EN(K) and the symmetric energy ES(K)