Rigidity theory for matroids

Mike Develin, Jeremy L. Martin, Victor S Reiner

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationFPSAC Proceedings 2005 - 17th Annual International Conference on Formal Power Series and Algebraic Combinatorics
Pages599-610
Number of pages12
StatePublished - Dec 1 2005
Event17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05 - Taormina, Italy
Duration: Jun 20 2005Jun 25 2005

Other

Other17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05
Country/TerritoryItaly
CityTaormina
Period6/20/056/25/05

Keywords

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

Fingerprint

Dive into the research topics of 'Rigidity theory for matroids'. Together they form a unique fingerprint.

Cite this