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

RWTH Publications (RWTH Aachen)

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 $656500 \pm 2500$. 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.

Authors 5

1. Jason Cantarella

   University of Georgia

   Affiliation as printed

   Mathematics Department , University of Georgia , Athens , GA , USA

2. Tetsuo Deguchi

   Ochanomizu University

   Affiliation as printed

   Department of Physics , Ochanomizu University , Bunkyo-ku , Tokyo , Japan

3. H. Ralph Schumacher Aachen Institut für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Mathematik , RWTH Aachen University , Aachen , Germany

4. Clayton Shonkwiler

   Colorado State University

   Affiliation as printed

   Department of Mathematics , Colorado State University , Fort Collins , CO , USA

5. Erica Uehara

   Kyoto University · Kyoto College of Graduate Studies for Informatics

   Affiliation as printed

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