Dehn–Thurston coordinates for SL3 basis webs in closed surfaces

The S L 3 -skein algebra of a closed surface Σ 𝑔 is a quantization of the S L 3 character variety of Σ 𝑔 , where g denotes the genus of the surface. This algebra admits a basis consisting of non-elliptic web diagrams in Σ 𝑔 after Sikora–Westbury [22] . In this paper, we introduce explicit S L 3 version of Dehn–Thurston coordinates for non-elliptic web diagrams on Σ 𝑔 , yielding a parametrization by a submonoid of ℤ 𝑑 . The idea is to tropicalize the Fenchel–Nielsen coordinates for the S L 3 -Hitchin component [23] . Here 𝑑 = 1 6 ⁢ 𝑔 − 1 6 for 𝑔 ≥ 2 and 𝑑 = 4 in the torus case 𝑔 = 1 , coinciding with the dimension of the corresponding character variety.