Biased Graphs. VI. Synthetic Geometry

TL;DR

This paper develops a synthetic geometric approach to represent biased graph matroids, extending previous algebraic methods and providing a more general framework for understanding their embeddings in projective spaces.

Contribution

It introduces a synthetic geometric method for representing biased graph matroids, broadening the scope beyond algebraic techniques and previous work.

Findings

Synthetic geometry effectively represents biased graph matroids.

The approach generalizes previous algebraic representations.

Provides new insights into embeddings in projective spaces.

Abstract

A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called balanced, such that no theta subgraph contains exactly two balanced circles. A biased graph $\Omega$ has two natural matroids, the frame matroid $G(\Omega)$, and the lift matroid $L(\Omega)$, and their extensions the full frame matroid $G^{{}^{{}_{{}_\bullet}}\!}(\Omega)$ and the extended (or complete) lift matroid $L_0(\Omega)$. In Part IV we used algebra to study the representations of these matroids by vectors over a skew field and the corresponding embeddings in Desarguesian projective spaces. Here we redevelop those representations, independently of Part IV and in greater generality, by using synthetic geometry.