Patchworking Oriented Matroids

TL;DR

This paper introduces a topological representation of oriented matroids derived from triangulations of simplices using a patchworking method, linking combinatorial and tropical geometric structures.

Contribution

It develops a novel patchworking technique to directly obtain topological representations of oriented matroids from polyhedral triangulations.

Findings

Derived a topological model from polyhedral structures

Rephrased patchworking as a controlled cell merging process

Proposed a new technique to prove regularity of cell complexes

Abstract

In a previous work, we gave a construction of (not necessarily realizable) oriented matroids from a triangulation of a product of two simplices. In this follow-up paper, we use a variant of Viro's patchworking to derive a topological representation of the oriented matroid directly from the polyhedral structure of the triangulation, hence finding a combinatorial manifestation of patchworking besides tropical algebraic geometry. We achieve this by rephrasing the patchworking procedure as a controlled cell merging process, guided by the structure of tropical oriented matroids. A key insight is a new promising technique to show that the final cell complex is regular.