Coordinates for ${\mathrm{S}\mathrm{L}}_3$-web basis elements in closed surfaces

The ${\mathrm{S}\mathrm{L}}_3$-skein algebra of a closed surface ${\mathrm{\Sigma }}_g$ is a quantization of the ${\mathrm{S}\mathrm{L}}_3$ character variety of ${\mathrm{\Sigma }}_g$, where $g$ denotes the genus of the surface. This algebra admits a basis consisting of non-elliptic web diagrams in ${\mathrm{\Sigma }}_g$. In this paper, we introduce explicit coordinates for non-elliptic web diagrams on ${\mathrm{\Sigma }}_g$, 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.