Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings

Ilke Çanakçi, Ralf Schiffler*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review


We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras and that are interesting on their own right as combinatorially defined rings. The elements of these rings are residue classes of unions of certain labeled graphs that were instrumental to construct canonical bases in the theory of cluster algebras. We obtain several rings by varying the conditions on the structure as well as the labeling of the graphs. A general form of this ring contains all cluster algebras of unpunctured surface type. The definition of the rings requires the snake graph calculus which we also complete in this article building on two earlier articles on the subject. Identities in the snake ring correspond to bijections between the posets of perfect matchings of the graphs. One of the main results of this article is the completion of the explicit construction of these bijections.

Original languageEnglish
Pages (from-to)1145-1226
Number of pages82
JournalInternational Mathematics Research Notices
Issue number4
Early online date17 Jul 2017
Publication statusPublished - Feb 2019
Externally publishedYes


The authors were supported by NSF grant DMS-10001637; I.C. was also supported by EPSRC grant number EP/K026364/1, UK, and by the University of Leicester; and R.S. was also supported by the NSF grants DMS-1254567, DMS-1101377, and by the University of Connecticut.

FundersFunder number
National Science FoundationDMS-10001637
Directorate for Mathematical and Physical Sciences1254567, 1101377
University of Connecticut
Engineering and Physical Sciences Research CouncilEP/K026364/1
University of LeicesterDMS-1254567, DMS-1101377


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