Random knotting in very long off-lattice self-avoiding polygons
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910224205479936 |
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| author | Cantarella, Jason Deguchi, Tetsuo Schumacher, Henrik Shonkwiler, Clayton Uehara, Erica |
| author_facet | Cantarella, Jason Deguchi, Tetsuo Schumacher, Henrik Shonkwiler, Clayton Uehara, Erica |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_04102 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Random knotting in very long off-lattice self-avoiding polygons Cantarella, Jason Deguchi, Tetsuo Schumacher, Henrik Shonkwiler, Clayton Uehara, Erica Statistical Mechanics Geometric Topology Probability 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. |
| title | Random knotting in very long off-lattice self-avoiding polygons |
| topic | Statistical Mechanics Geometric Topology Probability |
| url | https://arxiv.org/abs/2601.04102 |