Triangulations, order polytopes, and generalized snake posets

TL;DR

This paper studies the geometric and combinatorial properties of order polytopes from generalized snake posets, revealing their volume extremities, unimodular triangulations, flip counts, and symmetries.

Contribution

It characterizes volume extremities, circuits, and flip graphs of order polytopes from generalized snake posets, introducing twists that preserve regular triangulations.

Findings

Identified minimal and maximal volume order polytopes among generalized snake posets.

Proved all triangulations of these polytopes are unimodular.

Connected flip graphs to Cayley graphs of symmetric groups.

Abstract

This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in these order polytopes and then conclude that every triangulation is unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice of filters comes from a ladder is the Cayley graph of a symmetric group. Lastly, we introduce an operation on triangulations called twists and prove that twists preserve regular triangulations.