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

Non-contractible loops in the dense O(n) loop model on the cylinder