# A universality theorem for stressable graphs in the plane

## TL;DR

This paper proves a universality theorem for planar graphs with a given stress-oriented matroid, showing the complexity of their realization spaces and connecting to stratifications in algebraic geometry and tensegrity configurations.

## Contribution

It introduces a universality theorem for stressable graphs in the plane with prescribed stress-oriented matroids, linking combinatorics, geometry, and algebraic stratifications.

## Key findings

- Realization spaces can be arbitrarily complicated.
- Established a universality theorem for stressable graphs.
- Connected stress configurations to stratifications of Grassmannians.

## Abstract

Universality theorems (in the sense of N. Mn\"{e}v) claim that the realization space of a combinatorial object (a point configuration, a hyperplane arrangement, a convex polytope, etc.) can be arbitrarily complicated. In the paper, we prove a universality theorem for a graph in the plane with a prescribed \textit{oriented matroid of stresses}, that is the collection of signs of all possible equilibrium stresses of the graph. This research is motivated by the Grassmanian stratification (Gelfand, Goresky, MacPherson, Serganova) by thin Schubert cells, and by a recent series of papers on stratifications of configuration spaces of tensegrities (Doray, Karpenkov, Schepers, Servatius).