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arXiv · 1410.3035

Laminations from the symplectic double

Dylan G. L. Allegretti
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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.

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Dylan G. L. Allegretti. 2016-11-26. Laminations from the symplectic double. https://doi.org/10.1007/s10711-018-0339-0

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