Knot complexity and the probability of random knotting

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 aspect ratio $p$ of the tube length to the diameter of the {\it ideal knot} of $K$, which is a tubular representation of $K$ in its maximally inflated state.