Knot probabilities in equilateral random polygons

Research output: Contribution to journalArticlepeer-review


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

Colleges, School and Institutes


We consider the probability of knotting in equilateral random polygons in Euclidean 3-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 Sep 2021