Flows on graphs with cycles, locally gentle algebras, and the Mutoperhedron

Abstract

Flow cones of a directed acyclic graph admit a family of unimodular triangulations given by Danilov, Karzanov, and Koshevoy (DKK) whose normal fans are related to (generalizations) of the associahedron and permutahedron. A correspondence between these triangulations for certain graphs and maximal cones of a $g$-vector fan of a gentle quiver associated to the graph was discovered by von Bell, Braun, Bruegge, Hanely, Peterson, Serhiyenko, and Yip in 2022. This correspondence has been fruitful in uncovering lattice structures in the triangulations. We start by showing that this correspondence is actually a linear isomorphism. We then consider flow cones of certain graphs with cycles. For this case, we give a DKK-like triangulation of the cone, and extend the correspondence to the finite $g$-vector fan of a corresponding locally gentle quiver. In addition, we extend to cyclic graphs a mysterious result of Postnikov--Stanley and Baldoni--Vergne, giving the volume of flow polytopes of acyclic graphs as the number of certain integer flows on the same graph. We illustrate our results with a two-parameter family of cyclic graphs that includes a cycle graph and nested 2-cycles as special cases. We show that the fans of its DKK-like triangulations are respectively isomorphic to the normal fan of the cyclohedron and of a new polytope with the same $f$-vector but different combinatorial type than the permutohedron, which we call the mutoperhedron.