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.
Highlights
The tendency of long random filaments to become knotted is familiar to everyone carrying headphone cables in their pocket
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
Distinct kinds of knot are classified (the simplest examples are shown in figure 1(a)), and so we can ask what the probabilities of different knot types are in closed random walks
Summary
The tendency of long random filaments to become knotted is familiar to everyone carrying headphone cables in their pocket. It seems natural to expect that the probability that a random closed curve in three dimensions is knotted increases with its length. Random knotting—especially in closed random walks—has been studied at least since the 1960s. It was conjectured [1, 2] that sufficiently long linear polymers in dilute solution, undergoing a ring closure reaction, would produce knotted ring polymers with high probability. Closed random walks of sufficient length must be generated, whose knotting is analysed to investigate the asymptotics. Distinct kinds of knot are classified (the simplest examples are shown in figure 1(a)), and so we can ask what the probabilities of different knot types are in closed random walks
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