Equatorial Flow Triangulations of Gorenstein Flow Polytopes

Generalizing work of Athanasiadis for the Birkhoff polytope and Reiner and Welker for order polytopes, in 2007 Bruns and Römer proved that any Gorenstein lattice polytope with a regular unimodular triangulation admits a regular unimodular triangulation that is the join of a special simplex with a triangulated sphere. These are sometimes referred to as equatorial triangulations. We apply these techniques to give purely combinatorial descriptions of previously-unstudied triangulations of Gorensten flow polytopes. Further, we prove that the resulting equatorial flow polytope triangulations are usually distinct from the family of triangulations obtained by Danilov, Karzanov, and Koshevoy via framings. We find the facet description of the reflexive polytope obtained by projecting a Gorenstein flow polytope along a special simplex. Finally, we show that when a partially ordered set is strongly planar, equatorial triangulations of a related flow polytope can be used to produce new unimodular triangulations of the corresponding order polytope.