Unbounded š¯”°š¯”©3-laminations and their shear coordinates

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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 sl 3 ; ā€  , which we call unbounded sl 3-laminations, based on Kuperberg's sl 3-webs. We introduce their tropical cluster coordinates as an sl 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 ; ā€  of Goncharov and Shen, we describe a geometric gluing procedure of unbounded sl 3-laminations with pinnings via "shearings". We also investigate a relation to the graphical basis of the sl 3-skein algebra of Ishibashi and Yuasa (2023), which conjecturally leads to a quantum duality map.

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