Hard Squares with Negative Activity on Cylinders with Odd Circumference

• Let C-m,C-n be the graph on the vertex set {1, ..., m} x {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 C-m,C-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 Sigma(m,n) of independent sets in C-m,C-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 Sigma(m,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 Sigma(m,n) coincides with minus the partition function of the corresponding hard square model with activity -1.

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