Painted Tropical Complexes

TL;DR

This paper introduces painted tropical complexes, establishes a poset structure on them, and demonstrates that multiplihedra are special cases of secondary polytopes, linking tropical geometry with polytope theory.

Contribution

It defines painted tropical complexes, describes their poset structure, and proves that multiplihedra are secondary polytopes, connecting tropical geometry with polytope combinatorics.

Findings

Poset structure on painted tropical complexes

Equivalence of this poset to face lattice of a secondary polytope

Multiplihedra are shown to be secondary polytopes

Abstract

We define the notion of a painted tropical $A$-complex and describe a poset structure on the set of all such complexes. This poset is equivalent to the face lattice of a secondary polytope $\Sigma (\bar{A}_\alpha )$ where $\bar{A}_\alpha$ is built from $A$ and an additional point $\alpha$. As a central application, we show that multiplihedra are also secondary polytopes.