  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,
$ \sim \exp (-N/N_0)$, where $N_0$ marks the crossover from a mostly unknotted
(ie topologically simple) to a mostly knotted (ie topologically complex)
ensemble. In the present work we use computational simulation to look closer
into the variation of $N_0$ for a variety of polymer models. Among models
examined, $N_0$ 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.
