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

Journal of Physics A Mathematical and Theoretical, vol. 59, pp. 145205

Abstract

Abstract 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 656 500 ± 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.

Authors 4

1. Jason Cantarella

   University of Georgia

   Affiliation as printed

   Mathematics Department, University of Georgia, Mathematics Department, Athens, 30602, United States

2. Tetsuo Deguchi

   Ochanomizu University

   Affiliation as printed

   Department of Physics Faculty of Science, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Bunkyo, Tokyo, 112-8610, Japan

3. Clayton Shonkwiler

   Colorado State University

   Affiliation as printed

   Mathematics, Colorado State University, Department of Mathematics, Colorado State University, Campus Delivery 1874, Fort Collins, Colorado, 80523, United States

4. Erica Uehara

   Kyoto University

   Affiliation as printed

   Graduate School of Informatics, Kyoto University, Graduate School of Informatics, Sakyo-ku, Kyoto, 606-8501, Japan