  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 $m \times
n$ rectangular grid on the square lattice. For $m=2$, $p_{2 \times n} =
2^{n-1}-1$. And, for integers $m,n \geq 3$, $$2^{m+n-3}
\left(\frac{17}{10}\right)^{(m-2)(n-2)} \ \leq \ p_{m \times n} \ \leq \
2^{m+n-3} \left(\frac{31}{16}\right)^{(m-2)(n-2)}.$$
