Quiver combinatorics and triangulations of cyclic polytopes

Motivated by higher homological algebra, we associate quivers to triangulations of even-dimensional cyclic polytopes and prove two results showing what information about the triangulation is encoded in the quiver. We first show that the cut quivers of Iyama and Oppermann correspond precisely to 2 d -dimensional triangulations without interior ( d + 1)- simplices. This implies that these triangulations form a connected subgraph of the flip graph. Our second result shows how the quiver of a triangulation can be used to identify mutable internal d -simplices. This points towards what a theory of higher-dimensional quiver mutation might look like and gives a new way of understanding flips of triangulations of even-dimensional cyclic polytopes.