Monte Carlo results for projected self-avoiding polygons: a two-dimensional model for knotted polymers

We introduce a two-dimensional lattice model for the description of knotted polymer rings. A polymer configuration is modelled by a closed polygon drawn on the square diagonal lattice. with possible crossings describing pairs of strands of polymer passing on top of each other. Each polygon configuration can be viewed as the two-dimensional projection of a particular knot. We study numerically the statistics of large polygons with a fixed knot type, using a generalization of the BFACF algorithm for self-avoiding walks. This new algorithm incorporates both the displacement of crossings and the three types of Reidemeister transformations preserving the knot topology. Its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favour of this ergodicity are given together with a tentative sketch of proof. Assuming this ergodicity, we obtain numerically the following results for the statistics of knotted polygons: in the limit of a low crossing fugacity, we find a localization along the polygon of all the primary factors forming the knot. Increasing the crossing fugacity gives rise to a transition from a self-avoiding walk to a branched polymer behaviour.