## Abstract

Combinatorial rigidity theory seeks to describe the rigidity or exibility of bar-joint frameworks in Rd in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity theory by replacing G with an arbitrary representable matroid M. The notions of rigidity independence and parallel independence, as well as Laman's and Recski's combinatorial characterizations of 2-dimensional rigidity for graphs, can naturally be extended to this wider setting. As we explain, many of these fundamental concepts really depend only on the matroid associated with G (or its Tutte polynomial), and have little to do with the special nature of graphic matroids or the field ℝ. Our main result is a "nesting theorem" relating the various kinds of independence. Immediate corollaries include generalizations of Laman's Theorem, as well as the equality of 2-rigidity independence and 2-parallel independence. A key tool in our study is the space of photos of M, a natural algebraic variety whose irreducibility is closely related to the notions of rigidity independence and parallel independence. The number of points of this variety, when working over a finite field, turns out to be an interesting Tutte polynomial evaluation.

Original language | English (US) |
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Title of host publication | FPSAC Proceedings 2005 - 17th Annual International Conference on Formal Power Series and Algebraic Combinatorics |

Pages | 599-610 |

Number of pages | 12 |

State | Published - Dec 1 2005 |

Event | 17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05 - Taormina, Italy Duration: Jun 20 2005 → Jun 25 2005 |

### Other

Other | 17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05 |
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Country/Territory | Italy |

City | Taormina |

Period | 6/20/05 → 6/25/05 |

## Keywords

- Combinatorial rigidity
- Laman's Theorem
- Matroid
- Parallel redrawing
- Tutte polynomial