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Recovering the topology of surfaces from cluster algebras

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Abstract

We present an effective method for recovering the topology of a bordered oriented surface with marked points from its cluster algebra. The information is extracted from the maximal triangulations of the surface, those that have exchange quivers with maximal number of arrows in the mutation class. The method gives new proofs of the automorphism and isomorphism problems for the surface cluster algebras as well as the uniqueness of the Fomin–Shapiro–Thurston block decompositions of the exchange quivers of the surface cluster algebras. The previous proofs of these results followed a different approach based on Gu’s direct proof of the last result. The method also explains the exceptions to these results due to pathological problems with the maximal triangulations of several surfaces.

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Acknowledgements

We are grateful to Sergey Fomin and Misha Shapiro for their very helpful suggestions and comments on the first version of the paper. We are indebted to the referee for pointing out inaccuracies and making many suggestions which improved the paper. M. Y. would like to thank Newcastle University and the Max Planck Institute for Mathematics in Bonn for the warm hospitality during visits in the Fall of 2015.

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Eric Bucher

  2. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Milen Yakimov

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  1. Eric Bucher
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  2. Milen Yakimov
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Correspondence to Milen Yakimov.

Additional information

The research of E.B. has been supported by a GAANN fellowship and a VIGRE fellowship through the NSF Grant DMS-0739382 and that of M.Y. by the NSF Grants DMS-1303038 and DMS-1601862.

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Bucher, E., Yakimov, M. Recovering the topology of surfaces from cluster algebras. Math. Z. 288, 565–594 (2018). https://doi.org/10.1007/s00209-017-1901-4

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