On the Symmetric Molecular Conjectures

Porta, Josep M.; Ros, Lluís; Schulze, Bernd; Sljoka, Adnan; Whiteley, Walter

doi:10.1007/978-94-007-7214-4_20

Cited by 5 publications

(1 citation statement)

References 18 publications

(24 reference statements)

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“…We note that for finite molecular frameworks, their generic (local) rigidity was recently characterised by the celebrated result of Katoh and Tanigawa [14]. However, their generic global rigidity has not yet been characterised, and there are also no generic local or global rigidity characterisations for infinite L-periodic or finite symmetric molecular frameworks [18].…”

Section: Global Rigidity Of Body-bar Frameworkmentioning

confidence: 87%

Sufficient Conditions for the Global Rigidity of Periodic Graphs

Kaszanitzky

Király

Schulze

2021

Discrete Comput Geom

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Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension $$d>2$$ d > 2 . We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in $${\mathbb {R}}^d$$ R d on at most d vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in $${\mathbb {R}}^{2d}$$ R 2 d to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite body-bar frameworks.

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Section: Global Rigidity Of Body-bar Frameworkmentioning

confidence: 87%

Sufficient Conditions for the Global Rigidity of Periodic Graphs

Kaszanitzky

Király

Schulze

2021

Discrete Comput Geom

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Conclusions

Bohigas¹,

Manubens²,

Ros³

2016

Singularities of Robot Mechanisms

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Linking rigid bodies symmetrically

Schulze

Tanigawa

2014

European Journal of Combinatorics

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The mathematical theory of rigidity of body-bar and body-hinge frameworks provides a useful tool for analyzing the rigidity and flexibility of many articulated structures appearing in engineering, robotics and biochemistry. In this paper we develop a symmetric extension of this theory which permits a rigidity analysis of body-bar and body-hinge structures with point group symmetries.The infinitesimal rigidity of body-bar frameworks can naturally be formulated in the language of the exterior (or Grassmann) algebra. Using this algebraic formulation, we derive symmetry-adapted rigidity matrices to analyze the infinitesimal rigidity of body-bar frameworks with Abelian point group symmetries in an arbitrary dimension. In particular, from the patterns of these new matrices, we derive combinatorial characterizations of infinitesimally rigid body-bar frameworks which are generic with respect to a point group of the form Z/2Z × · · · × Z/2Z. Our characterizations are given in terms of packings of bases of signed-graphic matroids on quotient graphs. Finally, we also extend our methods and results to body-hinge frameworks with Abelian point group symmetries in an arbitrary dimension. As special cases of these results, we obtain combinatorial characterizations of infinitesimally rigid body-hinge frameworks with C 2 or D 2 symmetry -the most common symmetry groups found in proteins.

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On the Symmetric Molecular Conjectures

Cited by 5 publications

References 18 publications

Sufficient Conditions for the Global Rigidity of Periodic Graphs

Sufficient Conditions for the Global Rigidity of Periodic Graphs

Conclusions

Linking rigid bodies symmetrically

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