Triangulations of Flow Polytopes

ABSTRACT Given a polytope P, its secondary polytope has vertices corresponding to the regular tri- angulations of P, and two triangulations are connected by an edge in the secondary polytope if they di?er by a ?ip. We propose to study the secondary polytope of a class of polytopes called ?ow polytopes. Danilov, Karzanov and Koshevoy (DKK) have devised a method for constructing regular unimodular triangulations of ?ow polytopes with unit ?ow. In some of our previous work, we have shown that for a particular class of ?ow polytopes with Catalan volume, the DKK triangulations have dual graphs which are Tamari lattices and lattices of subsets of the Type A root posets. No general method for constructing triangulations of ?ow polytopes with non-unit ?ow is known. We are eager to develop theory along these lines, as there are classes of ?ow polytopes with non-unit ?ow with combinatorially-interesting volumes (involving numbers of parking functions), and it is of interest to study the triangulation space of these polytopes. 1