  We introduce a two-dimensional lattice model for the description of knotted
polymer rings. A polymer configuration is modeled 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
favor 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 behavior.
