Knotting in stretched polygons

The knotting in a lattice polygon model of ring polymers is examined when a stretching force is applied to the polygon. By examining the incidence of cut-planes in the polygon, we prove a pattern theorem in the stretching regime for large applied forces. This theorem can be used to examine the incidence of entanglements such as knotting and writhing. In particular, we prove that for arbitrarily large positive, but finite, values of the stretching force, the probability that a stretched polygon is knotted approaches 1 as the length of the polygon increases. In the case ofwrithing, we prove that for stretched polygons of length n, and for every function f(n) = o(root n), the probability that the absolute value of the mean writhe is less than f(n) approaches 0 as n -> 8, for sufficiently large values of the applied stretching force.