Mean unknotting times of random knots and embeddings

Abstract

We study mean unknotting times of knots and knot embeddings by crossing reversals, in aproblem motivated by DNA entanglement. Using self-avoiding polygons (SAPs) andself-avoiding polygon trails (SAPTs) we prove that the mean unknotting time growsexponentially in the length of the SAPT and at least exponentially with the lengthof the SAP. The proof uses Kesten’s pattern theorem, together with results formean first-passage times in the two-parameter Ehrenfest urn model. We use thepivot algorithm to generate random SAPTs of up to 3000 steps and calculate thecorresponding unknotting times, and find that the mean unknotting time grows veryslowly, even at moderate lengths. Our methods are quite general—for example,the lower bound on the mean unknotting time applies also to Gaussian randompolygons.