On Random Knots
In this paper, we consider knotting of Gaussian random polygons in 3-space. A Gaussian random polygon is a piecewise linear circle with n edges in which the length of the edges follows a Gaussian distribution. We prove a continuum version of Kesten's Pattern Theorem for these polygons, and use this to prove that the probability that a Gaussian random polygon of n edges in 3-space is knotted tends to one exponentially rapidly as n tends to infinity. We study the properties of Gaussian random knots, and prove that the entanglement complexity of Gaussian random knots gets arbitrarily large as n tends to infinity. We also prove that almost all Gaussian random knots are chiral.
Diao, Y., Pippenger, N., & Sumners, D.W. (1994). On random knots. Journal of Knot Theory and Its Ramifications, 3(3), 419-429. doi:10.1142/S0218216594000307