Generic combinatorial rigidity of periodic frameworks
type: Item
instance of: scholarly article
title: Generic combinatorial rigidity of periodic frameworks
zbMATH Open document ID: 1268.52021
author: Justin Malestein
author: Louis Theran
published in: Advances in Mathematics
publication date: 2013-01-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.1837
review text: The authors give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks which is similar to the Maxwell-Laman theorem [\textit{G. Laman}, J. Eng. Math. 4, 331--340 (1970; Zbl 0213.51903)] from rigidity theory in the sense that it is formulated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms. The authors define a periodic framework as a triple \((\widetilde{G}, \varphi, \widetilde{\ell})\), where \(\widetilde{G}\) is a simple infinite graph; \(\varphi\) is a free \(\mathbb Z^2\)-action on \(\widetilde{G}\) by automorphisms such that the quotient is finite; and \(\widetilde{\ell}=(\widetilde{\ell_{ij}})\) assigns a length to each edge of \(\widetilde{G}\). They define a realization \(\widetilde{G}(p,L)\) of a periodic framework \((\widetilde{G}, \varphi, \widetilde{\ell})\) as a pair consisting of a mapping \(p\) of the vertex set \(V\widetilde(G)\) into \(\mathbb R^2\) and a representation \(\mathbb Z^2\to\mathbb R^2\) encoded by a \(2\times 2\)-matrix \(L\) such that: (1) the representation is equivariant with respect to the \(\mathbb Z^2\)-actions on \(\widetilde{G}\) and the plane; and (2) the specified edge lengths are preserved by \(p\). A realization \(\widetilde{G}(p,L)\) is called rigid if the only allowed continuous motions of \(p\) and \(L\) that preserve the action \(\varphi\) and the edge lengths are rigid motions of the plane. If \(\widetilde{G}(p,L)\) is rigid but ceases to be so if any \(\mathbb Z^2\)-orbit of edges in \(\widetilde{G}\) is removed it is minimally rigid. One of the main results reads as follows: A generic realization \(\widetilde{G}(p,L)\) of a generic periodic framework \((\widetilde{G}, \varphi, \widetilde{\ell})\) is minimally rigid if and only if its colored quotient graph is colored-Laman. The definitions of colored-Laman and colored quotient graphs are too technical to be given in this review, but we observe that (i) every c…
Mathematics Subject Classification ID: 52C25
Mathematics Subject Classification ID: 05C63
Mathematics Subject Classification ID: 05C75
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 6126022
zbMATH Keywords: combinatorial rigidity
zbMATH Keywords: matroid
zbMATH Keywords: periodic graph
reviewed by: Victor Alexandrov
MaRDI profile type: Publication
OpenAlex ID: W2076961106
arXiv ID: 1008.1837
DOI: 10.1016/J.AIM.2012.10.007
description: scientific article