Random Knotting: Theorems, Simulations and Applications

Abstract

This article describes some of the theoretical and simulation results on random entanglement, and give a few scientific applications. I will prove that, on the simple cubic lattice Z3, the probability that a randomly chosen n-edge polygon in Z3 is knotted goes to one exponentially rapidly with length n (Murphy’s Law of entanglement); in other words, all but exponentially few polygons of length n in Z3 are knotted. Measures of entanglement complexity of random knots and random arcs are discussed as well as application of random knotting to viral DNA packing.