Dimer statistics on the Klein bottle

The problem of enumerating close-packed dimers, or perfect matchings, on a quadratic lattice embedded on the Klein bottle is considered. Thomassen C. Thomassen, Tilings of the torus and the Klein Bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323(2) (1991) 605-635] characterized that there are six quadrilateral lattices embedded on the Klein bottle. Lu and Wu W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Phys. Lett. A 293 (2002) 235-246] had obtained a expression for the number of close-packed dimers on one of them. In this paper we investigate four other embeddings and obtain explicit expressions of the numbers of close-packed dimers and free energy per dimer by enumerating Pfaffians.