Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality

Abstract

In a companion paper (arXiv 2011.01768) we constructed non-negative integer coordinates $\Phi_\mathcal{T}$ for a distinguished collection $\mathcal{W}_{3, \widehat{S}}$ of $\mathrm{SL}_3$-webs on a finite-type punctured surface $\widehat{S}$, depending on an ideal triangulation $\mathcal{T}$ of $\widehat{S}$. We prove that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $\mathcal{T}^\prime$ is chosen then the coordinate change map relating $\Phi_\mathcal{T}$ and $\Phi_{\mathcal{T}^\prime}$ is a prescribed tropical cluster transformation. Moreover, when $\widehat{S}=\Box$ is an ideal square, we provide a topological geometric description of the Hilbert basis (in the sense of linear programming) of the non-negative integer cone $\Phi_\mathcal{T}(\mathcal{W}_{3, \Box}) \subset \mathbb{Z}_{\geq 0}^{12}$, and we prove that this cone canonically decomposes into 42 sectors corresponding topologically to 42 families of $\mathrm{SL}_3$-webs in the square.Comment: 39 pages, 27 figure