Knot probabilities in equilateral random polygons

Anda Xiong, Alexander J. Taylor, Mark Dennis, S G Whittington

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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 languageEnglish
Article number405001
Number of pages21
JournalJournal of Physics A: Mathematical and Theoretical
Issue number40
Publication statusPublished - 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).


  • 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)


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