## Abstract

We consider the probability of knotting in equilateral random polygons in Euclidean three-dimensional space, which model, for instance, random polymers. Results from an extensive Monte Carlo dataset of random polygons indicate a universal scaling formula for the knotting probability with the number of edges. This scaling formula involves an exponential function, independent of knot type, with a power law factor that depends on the number of prime components of the knot. The unknot, appearing as a composite knot with zero components, scales with a small negative power law, contrasting with previous studies that indicated a purely exponential scaling. The methodology incorporates several improvements over previous investigations: our random polygon data set is generated using a fast, unbiased algorithm, and knotting is detected using an optimised set of knot invariants based on the Alexander polynomial.

Original language | English |
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Article number | 405001 |

Number of pages | 21 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 54 |

Issue number | 40 |

DOIs | |

Publication status | Published - 9 Sept 2021 |

### Bibliographical note

Funding Information:This research was funded in part by the Leverhulme Trust Research Programme Grant No. RP2013-K-009, SPOCK: Scientific Properties of Complex Knots.

Publisher Copyright:

© 2021 The Author(s).

## Keywords

- Knots
- Knotting
- Polygons
- Random walks

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modelling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)