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Rigidity theory for matroids
Develin, Mike Martin, Jeremy L. Reiner, Victor
Develin, Mike
Martin, Jeremy L.
Reiner, Victor
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Abstract
Combinatorial rigidity theory seeks to describe the rigidity or flexibility 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 ideas 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 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 on this variety, when working over a finite field, turns out to be an interesting Tutte polynomial evaluation.
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This is the author's accepted manuscript.
Date
2007
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European Mathematical Society
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Citation
Rigidity theory for matroids (with Mike Develin and Victor Reiner), Commentarii Mathematici Helvetici 82 (2007), 197--233.