Bounds on multiple self-avoiding polygons

A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problem to this study, we consider multiple self-avoiding polygons in a confined region as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds for the number p_m×n of distinct multiple self-avoiding polygons in the m × n rectangular grid on the square lattice. For m = 2, p_2×n = 2^n-1 - 1.