Coordinates for ${\rm SL}_3$-web basis elements in closed surfaces

The ${\rm SL}_3$-skein algebra of a closed surface $\Sigma_g$ is a quantization of the ${\rm SL}_3$ character variety of $\Sigma_g$, where $g$ denotes the genus of the surface. This algebra admits a basis consisting of non-elliptic web diagrams in $\Sigma_g$. In this paper, we introduce explicit coordinates for non-elliptic web diagrams on $\Sigma_g$, yielding a parametrization by a submonoid of $\mathbb Z^{d}$. Here $d = 16g - 16$ for $g \ge 2$ and $d = 4$ in the torus case $g = 1$, coinciding with the dimension of the corresponding character variety.