Knot complexity and the probability of random knotting.

Abstract

The probability of a random polygon (or a ring polymer) having a knot type K should depend on the complexity of the knot K. Through computer simulation using knot invariants, we show that the knotting probability decreases exponentially with respect to knot complexity. Here we assume that some aspects of knot complexity are expressed by the minimal crossing number C and the "rope length" of K, which is defined by the smallest length of rope with unit diameter that can be tied to make the knot K.