Uniquely realizable crystalline structures

2025/10/22 by Dewar, Sean, Schulze, Bernd, Tanigawa, Shin-ichi +1
#14P99 (Secondary) #52C25 (Primary) 05C62 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2510.19529

Abstract

We construct infinite periodic versions of the stress matrix and establish sufficient conditions for periodic tensegrity frameworks to be globally rigid in ℝ^d in the cases when the lattice is either fixed, fully flexible, or flexible with a volume constraint for the fundamental domain. For the fixed and fully flexible lattice variants, we also establish necessary and sufficient conditions for generic infinite periodic bar-joint frameworks to be globally rigid in ℝ^d. These results provide periodic versions of the fundamental results of Connelly, as well as Gortler, Healy and Thurston on the global rigidity of generic finite bar-joint frameworks.

Citations

• Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach
• Sufficient conditions for the global rigidity of periodic graphs
• Global Rigidity of Periodic Graphs under Fixed-lattice Representations
• Liftings and stresses for planar periodic frameworks
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• Inductive constructions for frameworks on a two-dimensional fixed torus
• Generic rigidity with forced symmetry and sparse colored graphs
• The rigidity of periodic frameworks as graphs on a fixed torus
• Global Rigidity: The Effect of Coning
• Generic Global Rigidity

Related

• Global Rigidity of Periodic Graphs under Fixed-lattice Representations
• The almost periodic rigidity of crystallographic bar-joint frameworks
• Infinite Bar-Joint Frameworks, Crystals and Operator Theory
• Crystal Frameworks, Matrix-valued Functions and Rigidity Operators
• On affine motions and universal rigidity of tensegrity frameworks