Geodesic
Universal Geometric Coefficients for the Once-Punctured Torus
Séminaire lotharingien de combinatoire, Tome 71 (2014-2015)

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We construct universal geometric coefficients, over Z, Q, and R, for cluster algebras arising from the once-punctured torus. We verify that the once-punctured torus has a property called the Null Tangle Property. The universal geometric coefficients over Z and Q are then given by the shear coordinates of certain "allowable" curves in the torus. The universal geometric coefficients over R are given by the shear coordinates of allowable curves together with the normalized shear coordinates of certain other curves each of which is dense in the torus. We also construct the mutation fan for the once-punctured torus and recover a result of Nájera on g-vectors. 

@article{SLC_2014-2015_71_a4,
     author = {Nathan Reading},
     title = {Universal {Geometric} {Coefficients} for the {Once-Punctured} {Torus}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {71},
     year = {2014-2015},
     url = {http://geodesic.mathdoc.fr/item/SLC_2014-2015_71_a4/}
}
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Nathan Reading. Universal Geometric Coefficients for the Once-Punctured Torus. Séminaire lotharingien de combinatoire, Tome 71 (2014-2015). http://geodesic.mathdoc.fr/item/SLC_2014-2015_71_a4/