Abstract:
Let G_{m,n} be the graph on the vertex set {1, ..., m} \times {1, ..., n} in which there is an edge between (a,b) and (c,d) if and only if either (a,b)=(c,d\pm 1) or (a,b)=(c\pm 1, d), where the second index is computed modulo n. One may view G_{m,n} as a unit square grid on a cylinder with circumference n and height m. For odd n, we prove that the Euler characteristic of the simplicial complex \Sigma_{m,n} of independent sets in G_{m,n} is either 2 or -1, depending on whether or not gcd(m-1,n) is divisible by 3. The proof builds on previous work due to Johan 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.