Geodesic
Universal geometric coefficients for the four-punctured sphere (Extended Abstract)
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015).

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We construct universal geometric coefficients for the cluster algebra associated to the four-punctured sphere and obtain, as a by-product, the $g$ -vectors of cluster variables. We also construct the rational part of the mutation fan. These constructions rely on a classification of the allowable curves (the curves which can appear in quasi-laminations). The classification allows us to prove the Null Tangle Property for the four-punctured sphere, thus adding this surface to a short list of surfaces for which this property is known. The Null Tangle Property then implies that the shear coordinates of allowable curves are the universal coefficients. We compute these shear coordinates to obtain universal geometric coefficients. 
@article{DMTCS_2015_special_285_a65,
     author = {Barnard, Emily and Meehan, Emily and Polster, Shira and Reading, Nathan},
     title = {Universal geometric coefficients for the four-punctured sphere {(Extended} {Abstract)}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)},
     year = {2015},
     doi = {10.46298/dmtcs.2521},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2521/}
}
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Barnard, Emily; Meehan, Emily; Polster, Shira; Reading, Nathan. Universal geometric coefficients for the four-punctured sphere (Extended Abstract). Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015). doi : 10.46298/dmtcs.2521. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2521/

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