Toric Cubes

TL;DR

This paper explores the topological structure of toric cubes, which are semialgebraic sets defined by binomial inequalities, providing CW-complex decompositions and generalizing phylogenetics applications.

Contribution

It introduces explicit CW-complex decompositions of toric cubes, extending known results from phylogenetics to a broader mathematical context.

Findings

Constructed CW-complex decompositions of toric cubes

Identified boundary subcomplexes within these decompositions

Generalized phylogenetics space results to broader classes

Abstract

A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells are interiors of toric cubes and their boundaries are subcomplexes. The motivating example of a toric cube is the edge-product space in phylogenetics, and our work generalizes results known for that space.