Generic combinatorial rigidity of periodic frameworks

Abstract

We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms. To prove our rigidity theorem we introduce and develop periodic direction networks and Z2-graded-sparse colored graphs.

Some typographical errors fixed. 61 pages, to appear in Advances in Math

References in corpus (1)

• Generic rigidity of frameworks with orientation-preserving crystallographic symmetry

Cited by in corpus (27)

• Crystal frameworks, symmetry and affinely periodic flexes
• Generic rigidity of frameworks with orientation-preserving crystallographic symmetry
• The rigidity of periodic body-bar frameworks on the three-dimensional fixed torus
• Generic rigidity of reflection frameworks
• Frameworks with forced symmetry I: Reflections and rotations
• Matroids of Gain Graphs in Applied Discrete Geometry
• Rigid components in fixed-lattice and cone frameworks
• Periodic Rigidity on a Variable Torus Using Inductive Constructions
• A characterisation of generically rigid frameworks on surfaces of revolution
• Global Rigidity of Periodic Graphs under Fixed-lattice Representations
• Flexing infinite frameworks with applications to braced Penrose tilings
• Henneberg constructions and covers of cone-Laman graphs
• Polynomials for Crystal Frameworks and the Rigid Unit Mode Spectrum
• Ultrarigid periodic frameworks
• The rigidity of infinite graphs
• The first-order flexibility of a crystal framework
• Inductive constructions for frameworks on a two-dimensional fixed torus
• Finite motions from periodic frameworks with added symmetry
• Quotient graphs of symmetrically rigid frameworks
• Count Matroids of Group-Labeled Graphs
• Finite and infinitesimal rigidity with polyhedral norms
• Deformations of period lattices of flexible polyhedral surfaces
• Periodic body-and-bar frameworks
• Frameworks with forced symmetry II: Orientation-preserving crystallographic groups
• The almost periodic rigidity of crystallographic bar-joint frameworks
• Flexible placements of periodic graphs in the plane
• Symmetric contact systems of segments, pseudotriangulations and inductive constructions for corresponding surface graphs