Universal characteristics of polygonal knot probabilities

There is a striking qualitative similarity among the graphs of the relative probabilities of corresponding knot types across a wide range of random polygon models. In many cases one has theoretical results describing the asymptotic decay of these knot probabilities but, in the finite range, there is little theoretical knowledge and a variety of functional models have been used to fit the observed structures. In this paper we compare a selection of these models and study the extent to which each provides a successful fit for five distinct random knot models. One consequence of this study is that while such models are quite successful in this finite range, they do not provide the theoretically predicted asymptotic struc- ture. A second result is the observed similarity between the global knot probabilities and those arising from small perturbations of three ideal knots.