Laminations from the symplectic double

Abstract

Let $S$ be a compact oriented surface with boundary together with finitely many marked points on the boundary, and let $S^\circ$ be the same surface equipped with the opposite orientation. We consider the double $S_\mathcal{D}$ obtained by gluing the surfaces $S$ and $S^\circ$ along corresponding boundary components. We define a notion of lamination on the double and construct coordinates on the space of all such laminations. We show that this space of laminations is a tropical version of the symplectic double introduced by Fock and Goncharov. There is a canonical pairing between our laminations and the positive real points of the symplectic double. We derive an explicit formula for this pairing using the $F$-polynomials of Fomin and Zelevinsky.