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{\bf Jakob Jonsson}
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{\bf Hard Squares with Negative Activity on Cylinders with Odd Circumference}
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Let $C_{m,n}$ be the graph on the vertex set $\{1, \ldots, m\} \times
\{0, \ldots, n-1\}$ 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
$C_{m,n}$ 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 $C_{m,n}$ 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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