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

Abstract

The ${\mathrm{SL}}_3$-skein algebra of a closed surface ${\Sigma }_g$ is a quantization of the ${\mathrm{SL}}_3$ 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 ${\Sigma }_g$ after Sikora--Westbury. In this paper, we introduce explicit ${\mathrm{SL}}_3$ version of Dehn--Thurston coordinates for non-elliptic web diagrams on ${\Sigma }_g$, yielding a parametrization by a submonoid of ${\mathbb{Z}}^d$. The idea is to tropicalize the Fenchel--Nielsen coordinates for the ${\mathrm{SL}}_3$-Hitchin component. 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.

arXiv comment30 pages. Accepted version, to appear in Advances in Mathematics