A discrete formulation for three-dimensional winding number

For a smooth map $g:X\to U(N)$, where $X$ is a three-dimensional, oriented, and closed manifold, the winding number is defined as $W_3=\frac{1}{24{\pi }^2}{\int }_X\mathrm{T}\mathrm{r}[(g^{-1}dg{)}^3]$. We present a discrete formulation to compute $W_3$ based on the concept of $\theta$-gaps. Our approach provides a robust scheme that is directly applicable even to systems with accidental or symmetry-enforced degeneracies. Furthermore, we define two versions of the discrete flux: a simple unmodified flux that is highly practical and almost always quantized for fine grids, and a modified flux that strictly ensures integer quantization.