F-Matrices of Cluster Algebras from Triangulated Surfaces


For a given marked surface (SM) and a fixed tagged triangulation T of (SM), we show that each tagged triangulation \(T'\) of (SM) is uniquely determined by the intersection numbers of tagged arcs of T and tagged arcs of \(T'\). As a consequence, each cluster in the cluster algebra \({{\,\mathrm{{\mathcal {A}}}\,}}(T)\) is uniquely determined by its F-matrix which is a new numerical invariant of the cluster introduced by Fujiwara and Gyoda.

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  1. 1.

    Note that this definition is slightly different from the “intersection number” \(({{\,\mathrm{\delta }\,}}| \epsilon )\) defined in [7, Definition 8.4]. The intersection numbers in this paper coincide with entries of f-vectors in cluster algebras, and ones in [7, Definition 8.4] coincide with entries of d-vectors (see a paragraph right after Theorem 1.4 and [7]). They are the same if (SM) has no puncture.

  2. 2.

    The lower part of the right hand side of (1.1) is known as the Heawood number. This number appears in the version of the four-color theorem for higher genus surface [24].

  3. 3.

    This example is also related to coloring problems of closed surfaces. It is the example that proves that we need at least 7 colors to properly cover a graph on the torus.


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The authors are grateful to Tomoki Nakanishi and Osamu Iyama for helpful advice. They also thank Futaba Fujie and Masakazu Tsuda for helpful comments. Authors are Research Fellows of Society for the Promotion of Science (JSPS). This work was supported by JSPS KAKENHI Grant number JP17J04270, JP20J12675.

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Correspondence to Yasuaki Gyoda.

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Gyoda, Y., Yurikusa, T. F-Matrices of Cluster Algebras from Triangulated Surfaces. Ann. Comb. (2020). https://doi.org/10.1007/s00026-020-00507-2

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  • Marked surface
  • Tagged triangulation
  • Intersection number
  • Cluster algebra
  • F-matrix