for a property of the spins that can also be described in terms of polygons.

Such a spin quantity has been studied recently in and can be thought of as an algebraic generalization of the usual continuum notion of vorticity. Consider a cubic lattice and orient links and plaquettes. Next place a spin variable $\sigma=\pm 1$ at each vertex. The discrete vorticity $\omega_p$ on a plaquette $p$ is defined as $$\begin{aligned}
\label{eq:omegap}
\omega_p = 
\frac{1}{2}
\sum_{(ij)\in \partial p}\epsilon_{ij}^p\frac{1-\sigma_i\sigma_j}{2}
\, ,\end{aligned}$$ where the sum is over links on the boundary of $p$ and $\epsilon_{ij}^p=1$ if the link is oriented as the plaquette, and $-1$ otherwise. We remark that $\omega_p=0,\pm 1$ and the choice of plaquette orientation has no effect on the $\mathbb{Z}_2$ quantity $(-1)^{\omega_p}$ that we consider below. For definiteness we choose orientations as in figure \[fig:conf\_omega0\].

An intuitive view on discrete vorticity comes form thinking of the spins $\sigma_i =+1,-1$ as the elements $0,1$ in $\mathbb{Z}_2$. Then $\omega_p$ appears as the discrete integral (i.e. a sum) around a plaquette over the discrete derivatives $\frac{1}{2}(1-\sigma_i\sigma_j) \in\mathbb{Z}_2$. Here it is important to interpret the discrete derivative as a variable in $\mathbb{Z}$ rather than in $\mathbb{Z}_2$, and hence this sum can be non-zero 