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 of 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 of the number $p_{m\times n}$ of distinct multiple self-avoiding polygons in the $m \times n$rectangulargridonthesquarelattice\mathrm{.}For$m=2$,$p_{2 \times n} = 2^{n-1}-1$\mathrm{.}And,forintegers$m,n \geq 3$,$$2^{m+n-3} \left(\frac{17}{10}\right)^{(m-2)(n-2)} \ \leq \ p_{m \times n} \ \leq
2^{m+n-3} \left(\frac{31}{16}\right)^{(m-2)(n-2)}.$$