Equatorial Flow Triangulations of Gorenstein Flow Polytopes

TL;DR

This paper explores equatorial triangulations of Gorenstein flow polytopes, providing combinatorial descriptions, comparing with existing triangulations, and linking to order polytopes for strongly planar posets.

Contribution

It introduces new combinatorial descriptions of Gorenstein flow polytope triangulations and compares them with existing methods, also connecting to order polytopes in strongly planar cases.

Findings

New combinatorial descriptions of Gorenstein flow polytope triangulations

Equatorial flow polytope triangulations are often distinct from known framings-based triangulations

Facet description of the reflexive polytope from Gorenstein flow polytopes

Abstract

Generalizing work of Athanasiadis for the Birkhoff polytope and Reiner and Welker for order polytopes, in 2007 Bruns and R\"omer 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.