Equatorial Flow Triangulations of Gorenstein Flow Polytopes

TL;DR

This work develops equatorial triangulations for Gorenstein flow polytopes by leveraging Bruns–Römer’s join framework and graph-theoretic route decompositions. It provides a purely combinatorial construction of equatorial flow triangulations, proves regular unimodularity, and shows these triangulations can diverge from the canonical DKK framings, particularly when inner vertices have high indegree. A reflexive projection of a Gorenstein flow polytope is constructed, with a detailed facet description tied to the equatorial complex. In the strongly planar setting, the authors establish an integral equivalence between equatorial flow triangulations and Reiner–Welker’s equatorial triangulation of order polytopes, yielding new unimodular triangulations for strongly planar posets and connecting flow and order polytope triangulations through a shared combinatorial framework.

Abstract

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.

Figures (12)

• Figure 1: An example of a framed DAG where the edges are linearly ordered by $1 < 2$ and an example of coherent routes forming a maximal clique. The bottom two routes are the exceptional routes for this framed DAG.
• Figure 2: Example of a DAG with degree equality.
• Figure 3: Example of adding new edges (in orange) to a DAG to satisfy degree equality.
• Figure 4: Example of a route decomposition for the DAG found in figure \ref{['fig:degreeequality']}.
• Figure 5: An example of deleting routes

Theorems & Definitions (109)

• Definition 2.1
• Definition 2.2
• Theorem 2.3: Danilov, Karzanov, and Koshevoy DKK
• Example 2.4
• Definition 2.5
• Definition 2.6
• Example 2.7
• Theorem 2.8
• Definition 2.9
• Theorem 2.10