Unbounded 𝔰𝔩3-laminations and their shear coordinates

Ishibashi, Tsukasa; Kano, Shunsuke

doi:10.2140/agt.2025.25.1433

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

Tsukasa Ishibashi and Shunsuke Kano

Algebraic & Geometric Topology 25 (2025) 1433–1500

DOI: 10.2140/agt.2025.25.1433

Abstract

Generalizing the work of Fock and Goncharov on rational unbounded laminations, we give a geometric model of the tropical points of the cluster variety $𝒳_{{𝔰 𝔩}_{3}, Σ}$ , which we call unbounded ${𝔰 𝔩}_{3}$ -laminations, based on Kuperberg’s ${𝔰 𝔩}_{3}$ -webs. We introduce their tropical cluster coordinates as an ${𝔰 𝔩}_{3}$ -analogue of Thurston’s shear coordinates associated with any ideal triangulation. As a tropical analogue of gluing morphisms among the moduli spaces $𝒫_{{PGL}_{3}, Σ}$ of Goncharov and Shen, we describe a geometric gluing procedure of unbounded ${𝔰 𝔩}_{3}$ -laminations with pinnings via “shearings”. We also investigate a relation to the graphical basis of the ${𝔰 𝔩}_{3}$ -skein algebra of Ishibashi and Yuasa (2023), which conjecturally leads to a quantum duality map.

Keywords

higher lamination, shear coordinates, cluster algebra, skein algebra

Mathematical Subject Classification

Primary: 13F60, 57K20, 57K31

References

Publication

Received: 24 September 2022

Revised: 24 January 2024

Accepted: 16 February 2024

Published: 20 June 2025

Authors

Tsukasa Ishibashi
Mathematical Institute Tohoku University Sendai Japan

Shunsuke Kano
Mathematical Science Center for Co-creative Society Tohoku University Sendai Japan

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Volume 25, issue 3 (2025)