Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates

Tsukasa Ishibashi, Shunsuke Kano

Published 2022-04-19Version 1

We study the space $\mathcal{L}_{\mathfrak{sl}_3}^x(\Sigma,\mathbb{Q})$ of rational unbounded $\mathfrak{sl}_3$-laminations on a marked surface $\Sigma$. We introduce an $\mathfrak{sl}_3$-analogue of the Thurston's shear coordinates associated with any decorated triangulation, which gives rise to a natural identification $\mathcal{L}_{\mathfrak{sl}_3}^x(\Sigma,\mathbb{Q}) \cong \mathcal{X}_{\mathfrak{sl}_3,\Sigma}^\mathrm{uf}(\mathbb{Q}^{\mathrm{trop}})$. We also introduce the space $\mathcal{L}^p(\Sigma,\mathbb{Q})$ of rational unbounded $\mathfrak{sl}_3$-laminations with pinnings, which possesses the frozen coordinates as well. Then we give a tropical anologue of the amalgamation maps [FG06b] between them, which is indeed a procedure of gluing $\mathfrak{sl}_3$-laminations with "shearings". We also investigate a relation to the graphical basis of the $\mathfrak{sl}_3$-skein algebra [IY21], which conjecturally leads to a quantum duality map.