  A lattice model of critical dense polymers $O(0)$ is considered for the
finite cylinder geometry. Due to the presence of non-contractible loops with a
fixed fugacity $\xi$, the model is a generalization of the critical dense
polymers solved by Pearce, Rasmussen and Villani. We found the free energy for
any height $N$ and circumference $L$ of the cylinder. The density $\rho$ of
non-contractible loops is found for $N \rightarrow \infty$ and large $L$. The
results are compared with those obtained for the anisotropic quantum chain with
twisted boundary conditions. Using the latter method we obtained $\rho$ for any
$O(n)$ model and an arbitrary fugacity.
