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Frameworks with forced symmetry II: orientation-preserving crystallographic groups

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Abstract

We give a combinatorial characterization of minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, under the constraint of forced symmetry. The main theorems are proved by extending the methods of the first paper in this sequence from groups generated by a single rotation to groups generated by translations and rotations. The proof makes use of new families of matroids and submodular functions defined on crystallographic groups.

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Notes

  1. The terminology of “\((2,2)\)” and “\((1,1)\)” comes from the fact that spanning trees of finite graphs are “\((1,1)\)-tight” in the sense of [10]. The \(\varGamma \)-\((1,1)\) graphs defined here are, in a sense made more precise in [16, Section 5.2], analogous to spanning trees.

  2. The reference [14] is an earlier version of [16].

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Acknowledgments

We thank Igor Rivin for encouraging us to take on this project and many productive discussions on the topic. Our initial work on this topic was part of a larger effort to understand the rigidity and flexibility of hypothetical zeolites, which is supported by NSF CDI-I grant DMR 0835586 to Rivin and M. M. J. Treacy. LT is funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 247029-SDModels. JM is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 226135.

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  1. Einstein Institute of Mathematics, Givat Ram, The Hebrew University, 91904 , Jerusalem, Israel

    Justin Malestein

  2. Institut für Mathematik, Freie Universität Berlin, 14195 , Berlin, Germany

    Louis Theran

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  1. Justin Malestein
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Correspondence to Louis Theran.

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Malestein, J., Theran, L. Frameworks with forced symmetry II: orientation-preserving crystallographic groups. Geom Dedicata 170, 219–262 (2014). https://doi.org/10.1007/s10711-013-9878-6

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