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

Abstract

We study the space L x sl3 (Σ, Q) of rational unbounded sl 3 -laminations on a marked surface Σ. We introduce an sl 3 -analogue of the Thurston's shear coordinates associated with any decorated triangulation, which gives rise to a natural identificationWe also introduce the space L p sl3 (Σ, Q) of rational unbounded sl 3 -laminations with pinnings, which possesses the frozen coordinates as well. Then we give a tropical anologue of the amalgamation maps [FG06b] between them, which is indeed a procedure of gluing sl 3 -laminations with "shearings". We also investigate a relation to the graphical basis of the sl 3 -skein algebra [IY21], which conjecturally leads to a quantum duality map.

Train track combinatorics and cluster algebras

“…On the other hand, the geometric model of higher measured laminations is suggested in the case g = sl 3 in [DS20] and [IK22] but these are only rational points of A sl 3 ,Σ (R trop ) and X sl 3 ,Σ (R trop ), respectively.…”