Laminations from the symplectic double
Dylan G. L. Allegretti · Geometriae Dedicata · 2018
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.