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

Abstract

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

30 pages. This is Version 2. In the revised version, we updated the definition of essential web diagrams in a pair of pants in Definition 4.1. We also added several examples in Figure 7 to illustrate Definition 4.1