The abundance of unknots in various models of polymer loops

A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of N segments follows a decaying exponential form, ∼exp(-N/N₀), where N₀ marks the crossover from a mostly unknotted (i.e., topologically simple) to a mostly knotted (i.e., topologically complex) ensemble. In the present work, we use computational simulation to look closer into the variation of N₀ for a variety of polymer models. Among models examined, N₀ is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian-distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power-law tail.