Unbounded 𝔰𝔩3-laminations and their shear coordinates

Ishibashi, Tsukasa¹ ; Kano, Shunsuke²

¹Mathematical Institute, Tohoku University, Sendai, Japan
²Mathematical Science Center for Co-creative Society, Tohoku University, Sendai, Japan

Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1433-1500

Résumé

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.

Détail

DOI : 10.2140/agt.2025.25.1433

Keywords: higher lamination, shear coordinates, cluster algebra, skein algebra

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