Random knotting in very long off-lattice self-avoiding polygons

We present experimental results on knotting in off-lattice self-avoiding polygons in the bead-chain model. Using Clisby's tree data structure and the scale-free pivot algorithm, for each $k$ between $10$ and $27$ we generated $2^{43-k}$ polygons of size $n=2^k$. Using a new knot diagram simplification and invariant-free knot classification code, we were able to determine the precise knot type of each polygon. The results show that the number of prime summands of knot type $K$ in a random $n$-gon is very well described by a Poisson distribution. We estimate the characteristic length of knotting as $656500 \pm 2500$. We use the count of summands for large $n$ to measure knotting rates and amplitude ratios of knot probabilities more accurately than previous experiments. Our calculations agree quite well with previous on-lattice computations, and support both knot localization and the knot entropy conjecture.