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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2012.14202 |
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Table of Contents:
- In a companion paper (arXiv 2011.01768), we constructed nonnegative integer coordinates $Φ_\mathscr{T}(\mathscr{W}_{3, \hat{S}}) \subset \mathbb{Z}_{\geq 0}^N$ for the collection $\mathscr{W}_{3, \hat{S}}$ of reduced $\mathrm{SL}_3$-webs on a finite-type punctured surface $\hat{S}$, depending on an ideal triangulation $\mathscr{T}$ of $\hat{S}$. We show that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $\mathscr{T}^\prime$ is chosen, then the coordinate change map relating $Φ_\mathscr{T}(\mathscr{W}_{3, \hat{S}})$ to $Φ_{\mathscr{T}^\prime}(\mathscr{W}_{3, \hat{S}})$ is a tropical $\mathcal{A}$-coordinate cluster transformation. We can therefore view the webs $\mathscr{W}_{3, \hat{S}}$ as a concrete topological model for the Fock-Goncharov-Shen positive integer tropical points $\mathcal{A}_{\mathrm{PGL}_3, \hat{S}}^+(\mathbb{Z}^t)$.