Geodesic
Snake Graphs from Triangulated Orbifolds
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: generalized cluster algebra, cluster algebra, orbifold, snake graph. 
@article{SIGMA_2020_16_a137,
     author = {Esther Banaian and Elizabeth Kelley},
     title = {Snake {Graphs} from {Triangulated} {Orbifolds}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a137/}
}
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Esther Banaian; Elizabeth Kelley. Snake Graphs from Triangulated Orbifolds. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a137/

[1] Banaian E., Kelley E., “Snake graphs and frieze patterns from orbifolds”, Sém. Lothar. Combin., 82B (2019), 88, 12 pp., arXiv: 2003.13872 | MR | Zbl

[2] Berenstein A., Fomin S., Zelevinsky A., “Cluster algebras. III Upper bounds and double Bruhat cells”, Duke Math. J., 126 (2005), 1–52, arXiv: math.RT/0305434 | DOI | MR | Zbl

[3] Caldero P., Reineke M., J. Pure Appl. Algebra, 212 (2008), 2369–2380, On the quiver Grassmannian in the acyclic case pp., arXiv: math.RT/0611074 | DOI | MR | Zbl

[4] Canakci I., Schiffler R., “Snake graph calculus and cluster algebras from surfaces”, J. Algebra, 382 (2013), 240–281, arXiv: 1209.4617 | DOI | MR | Zbl

[5] Canakci I., Tumarkin P., “Bases for cluster algebras from orbifolds with one marked point”, Algebr. Comb., 2 (2019), 355–365, arXiv: 1711.00446 | DOI | MR | Zbl

[6] Chekhov L., “Orbifold Riemann surfaces and geodesic algebras”, J. Phys. A: Math. Theor., 42 (2009), 304007, 32 pp., arXiv: 0911.0214 | DOI | MR | Zbl

[7] Chekhov L., Mazzocco M., “Isomonodromic deformations and twisted Yangians arising in Teichmüller theory”, Adv. Math., 226 (2011), 4731–4775, arXiv: 0909.5350 | DOI | MR | Zbl

[8] Chekhov L., Shapiro M., “Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables”, Int. Math. Res. Not., 2014 (2014), 2746–2772, arXiv: 1111.3963 | DOI | MR | Zbl

[9] Efimov A. I., Quantum cluster variables via vanishing cycles, arXiv: 1112.3601

[10] Felikson A., Shapiro M., Tumarkin P., “Cluster algebras and triangulated orbifolds”, Adv. Math., 231 (2012), 2953–3002, arXiv: 1111.3449 | DOI | MR | Zbl

[11] Felikson A., Shapiro M., Tumarkin P., “Cluster algebras of finite mutation type via unfoldings”, Int. Math. Res. Not., 2012 (2012), 1768–1804, arXiv: 1006.4276 | DOI | MR | Zbl

[12] Felikson A., Tumarkin P., “Bases for cluster algebras from orbifolds”, Adv. Math., 318 (2017), 191–232, arXiv: 1511.08023 | DOI | MR | Zbl

[13] Fock V. V., Goncharov A. B., “Dual Teichmüller and lamination spaces”, Handbook of Teichmüller Theory, v. I, IRMA Lect. Math. Theor. Phys., 11, Eur. Math. Soc., Zürich, 2007, 647–684, arXiv: math.DG/0510312 | DOI | MR | Zbl

[14] Fock V. V., Goncharov A. B., “Cluster ensembles, quantization and the dilogarithm”, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 865–930, arXiv: math.AG/0311245 | DOI | MR | Zbl

[15] Fomin S., Shapiro M., Thurston D., “Cluster algebras and triangulated surfaces. I Cluster complexes”, Acta Math., 201 (2008), 83–146, arXiv: math.RA/0608367 | DOI | MR | Zbl

[16] Fomin S., Thurston D., Cluster algebras and triangulated surfaces, v. II, Mem. Amer. Math. Soc., 255, Lambda lengths, 2018, v+97 pp., arXiv: 1210.5569 | DOI | MR

[17] Fomin S., Zelevinsky A., “Cluster algebras. I Foundations”, J. Amer. Math. Soc., 15 (2002), 497–529, arXiv: math.RT/0104151 | DOI | MR | Zbl

[18] Fomin S., Zelevinsky A., “The Laurent phenomenon”, Adv. in Appl. Math., 28 (2002), 119–144, arXiv: math.CO/0104241 | DOI | MR | Zbl

[19] Fomin S., Zelevinsky A., “Cluster algebras. II Finite type classification”, Invent. Math., 154 (2003), 63–121, arXiv: math.RA/0208229 | DOI | MR | Zbl

[20] Fomin S., Zelevinsky A., “Cluster algebras. IV Coefficients”, Compos. Math., 143 (2007), 112–164, arXiv: math.RA/0208229 | DOI | MR | Zbl

[21] Gleitz A. S., Musiker G., Private communication

[22] Gross M., Hacking P., Keel S., Kontsevich M., “Canonical bases for cluster algebras”, J. Amer. Math. Soc., 31 (2018), 497–608, arXiv: 1411.1394 | DOI | MR | Zbl

[23] Gunawan E., Musiker G., “$T$-path formula and atomic bases for cluster algebras of type $D$”, SIGMA, 11 (2015), 060, 46 pp., arXiv: 1409.3610 | DOI | MR | Zbl

[24] Holm T., Jørgensen P., “A $p$-angulated generalisation of Conway and Coxeter's theorem on frieze patterns”, Int. Math. Res. Not., 2020 (2020), 71–90, arXiv: 1709.09861 | DOI | MR

[25] Kimura Y., Qin F., “Graded quiver varieties, quantum cluster algebras and dual canonical basis”, Adv. Math., 262 (2014), 261–312, arXiv: 1205.2066 | DOI | MR | Zbl

[26] Labardini-Fragoso D., Velasco D., “On a family of Caldero–Chapoton algebras that have the Laurent phenomenon”, J. Algebra, 520 (2019), 90–135, arXiv: 1704.07921 | DOI | MR | Zbl

[27] Lang W., The field $\mathbb{Q}(2\cos(\pi/n))$, its Galois group and length ratios in the regular $n$-gon, arXiv: 1210.1018

[28] Lee K., Schiffler R., “Positivity for cluster algebras”, Ann. of Math., 182 (2015), 73–125, arXiv: 1306.2415 | DOI | MR | Zbl

[29] Musiker G., Schiffler R., “Cluster expansion formulas and perfect matchings”, J. Algebraic Combin., 32 (2010), 187–209, arXiv: 0810.3638 | DOI | MR | Zbl

[30] Musiker G., Schiffler R., Williams L., “Positivity for cluster algebras from surfaces”, Adv. Math., 227 (2011), 2241–2308, arXiv: 0906.0748 | DOI | MR | Zbl

[31] Musiker G., Schiffler R., Williams L., “Bases for cluster algebras from surfaces”, Compos. Math., 149 (2013), 217–263, arXiv: 1110.4364 | DOI | MR | Zbl

[32] Musiker G., Williams L., “Matrix formulae and skein relations for cluster algebras from surfaces”, Int. Math. Res. Not., 2013 (2013), 2891–2944, arXiv: 1108.3382 | DOI | MR | Zbl

[33] Nakajima H., “Quiver varieties and cluster algebras”, Kyoto J. Math., 51 (2011), 71–126, arXiv: 0905.0002 | DOI | MR | Zbl

[34] Nakanishi T., “Structure of seeds in generalized cluster algebras”, Pacific J. Math., 277 (2015), 201–217, arXiv: 1409.5967 | DOI | MR

[35] Nakanishi T., Rupel D., “Companion cluster algebras to a generalized cluster algebra”, Trav. Math., 24 (2016), 129–149, Fac. Sci. Technol. Commun. Univ. Luxemb., Luxembourg, arXiv: 1504.06758 | MR

[36] Nakanishi T., Zelevinsky A., “On tropical dualities in cluster algebras”, Algebraic Groups and Quantum Groups, Contemp. Math., 565, Amer. Math. Soc., Providence, RI, 2012, 217–226, arXiv: 1101.3736 | DOI | MR | Zbl

[37] Paquette C., Schiffler R., “Group actions on cluster algebras and cluster categories”, Adv. Math., 345 (2019), 161–221, arXiv: 1703.06174 | DOI | MR | Zbl

[38] Propp J., Lattice structure for orientations of graphs, arXiv: math.CO/0209005

[39] Qin F., “Quantum cluster variables via Serre polynomials”, J. Reine Angew. Math., 668 (2012), 149–190, arXiv: 1004.4171 | DOI | MR | Zbl

[40] Satake I., “On a generalization of the notion of manifold”, Proc. Nat. Acad. Sci. USA, 42 (1956), 359–363 | DOI | MR | Zbl

[41] Schiffler R., “On cluster algebras arising from unpunctured surfaces. II”, Adv. Math., 223 (2010), 1885–1923, arXiv: 0809.2593 | DOI | MR | Zbl

[42] Schiffler R., Thomas H., “On cluster algebras arising from unpunctured surfaces”, Int. Math. Res. Not., 2009 (2009), 3160–3189, arXiv: 0712.4131 | DOI | MR | Zbl

[43] Thurston W. P., “On the geometry and dynamics of diffeomorphisms of surfaces”, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417–431 | DOI | MR | Zbl

[44] Thurston W. P., Three-dimensional geometry and topology, v. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997 | MR | Zbl