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Rigidity theory for matroids
dc.contributor.author | Develin, Mike | |
dc.contributor.author | Martin, Jeremy L. | |
dc.contributor.author | Reiner, Victor | |
dc.date.accessioned | 2010-06-17T21:17:44Z | |
dc.date.available | 2010-06-17T21:17:44Z | |
dc.date.issued | 2007 | |
dc.identifier.citation | Rigidity theory for matroids (with Mike Develin and Victor Reiner), Commentarii Mathematici Helvetici 82 (2007), 197--233. | |
dc.identifier.uri | http://hdl.handle.net/1808/6358 | |
dc.description | This is the author's accepted manuscript. | |
dc.description.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. | |
dc.publisher | European Mathematical Society | |
dc.relation.hasversion | http://arxiv.org/abs/math.CO/0503050 | |
dc.title | Rigidity theory for matroids | |
dc.type | Article | |
kusw.kuauthor | Martin, Jeremy L. | |
kusw.kudepartment | Mathematics | |
kusw.oastatus | fullparticipation | |
dc.identifier.doi | 10.4171/CMH/89 | |
kusw.oaversion | Scholarly/refereed, author accepted manuscript | |
kusw.oapolicy | This item meets KU Open Access policy criteria. | |
dc.rights.accessrights | openAccess |