Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 138, 50 pages      arXiv:2003.13872      https://doi.org/10.3842/SIGMA.2020.138
Contribution to the Special Issue on Cluster Algebras

Snake Graphs from Triangulated Orbifolds

Esther Banaian and Elizabeth Kelley
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Received March 31, 2020, in final form December 08, 2020; Published online December 17, 2020

Abstract
We give an explicit combinatorial formula for the Laurent expansion of any arc or closed curve on an unpunctured triangulated orbifold. We do this by extending the snake graph construction of Musiker, Schiffler, and Williams to unpunctured orbifolds. In the case of an ordinary arc, this gives a combinatorial proof of positivity to the generalized cluster algebra from this orbifold.

Key words: generalized cluster algebra; cluster algebra; orbifold; snake graph.

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