Polygons in restricted geometries subjected to infinite forces

We consider self-avoiding polygons in a restricted geometry, namely an infinite L × M tube in $Z^3$. These polygons are subjected to a force f, parallel to the infinite axis of the tube. When $f>0$ the force stretches the polygons, while when $f<0$ the force is compressive. We obtain and prove the asymptotic form of the free energy in both limits $f\to \pm \infty$. We conjecture that the $f\to -\infty$ asymptote is the same as the limiting free energy of 'Hamiltonian' polygons, polygons which visit every vertex in a $L\times M\times N$ box. We investigate such polygons, and in particular use a transfer-matrix methodology to establish that the conjecture is true for some small tube sizes. (paper)