Hard Squares with Negative Activity on Cylinders with Odd Circumference

Let Cm,n be the graph on the vertex set {1, . . . ,m} × {0, . . . , n − 1} in which there is an edge between (a, b) and (c, d) if and only if either (a, b) = (c, d ± 1) or (a, b) = (c ± 1, d), where the second index is computed modulo n. One may view Cm,n as a unit square grid on a cylinder with circumference n units. For odd n, we prove that the Euler characteristic of the simplicial complex Σm,n of independent sets in Cm,n is either 2 or −1, depending on whether or not gcd(m−1, n) is divisble by 3. The proof relies heavily on previous work due to Thapper, who reduced the problem of computing the Euler characteristic of Σm,n to that of analyzing a certain subfamily of sets with attractive properties. The situation for even n remains unclear. In the language of statistical mechanics, the reduced Euler characteristic of Σm,n coincides with minus the partition function of the corresponding hard square model with activity −1.