Laminations from the symplectic double

TL;DR

This paper introduces a new lamination space on the symplectic double of a surface, providing explicit coordinates and a pairing with positive real points, extending Fock-Goncharov's framework.

Contribution

It defines a lamination space on the symplectic double and constructs explicit coordinates and pairings, connecting to Fomin-Zelevinsky's $F$-polynomials.

Findings

Lamination space is a tropical version of the symplectic double.

Constructed explicit coordinates on the lamination space.

Derived an explicit pairing formula using $F$-polynomials.

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.