Journal of statistical physics vol:115 issue:1-2 pages:681-700
We consider single ring polymers which are confined on a plane but maintain a fixed three-dimensional knotted topology. The equilibrium statistics of such systems is studied on the basis of a model on square lattice in which the configurations are represented by N-step polygons with a number of self-intersections restricted to the minimum compatible with the topology. This allows to define the size, s, of the flat knots and to study their localization properties. Due to the presence of both excluded volume and attractive interactions, the model undergoes a theta transition. Accurate Monte Carlo results show that, while in the high temperature swollen regime both prime and composite knot components are localized ([s](N) similar to N-t, with t = 0), in the low temperature, compact phase they are fully delocalized (t = 1). Right at the theta transition weak localization prevails (t = 0.44 +/- 0.02). Part of the results can be interpreted by taking into account a dominance of figure eight shapes for the coarse grained knotted polymer configurations, and by applying the scaling theory of polymer networks of fixed topology. In particular t = 3/7 can be conjectured as an exact exponent characterizing the weak knot localization at the theta point.