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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
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 𝒳𝔰𝔩3,Σ, which we call unbounded 𝔰𝔩3-laminations, based on Kuperberg’s 𝔰𝔩3-webs. We introduce their tropical cluster coordinates as an 𝔰𝔩3-analogue of Thurston’s shear coordinates associated with any ideal triangulation. As a tropical analogue of gluing morphisms among the moduli spaces 𝒫PGL 3,Σ of Goncharov and Shen, we describe a geometric gluing procedure of unbounded 𝔰𝔩3-laminations with pinnings via “shearings”. We also investigate a relation to the graphical basis of the 𝔰𝔩3-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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