Séminaire Lotharingien de Combinatoire, 95B.91 (2026), 12 pp.
Antoine Abram, Jose Bastidas, Benjamin Dequêne, Alejandro H. Morales, GaYee Park and Hugh Thomas
Flows on Cyclic Graphs, Complete Gentle Algebras, and the Mutoperhedron
Abstract.
The flow cone of a directed acyclic graph admits a family of unimodular triangulations, as described by Danilov, Karzanov, and Koshevoy (DKK), that are of active interest.
von Bell, Braun, Bruegge, Hanely, Peterson, Serhiyenko, and Yip recently discovered a correspondence between these triangulations for certain graphs and maximal cones of a g-vector fan of an associated gentle quiver. This correspondence has already been fruitful in uncovering lattice structures in the triangulations. In this extended abstract, we extend the correspondence to certain graphs with cycles. We also give a cyclic analogue of the results of Baldoni-Vergne and Postnikov-Stanley, expressing the volume of the flow polytope of acyclic graphs as the number of certain integer flows on that same graph. In addition, we introduce a new polytope, the \emph{mutoperhedron}, whose f-vector is equal to that of the permutohedron but which has a different combinatorial type.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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