No fractional rigidity orders occur before third order, and second-order prestress stability can be checked by explicit linear-algebra criteria, but a general third-order test exists only when the rigidity matrix has one-dimensional null space.
Sufficient conditions for the extendibility of an n-th order flex of polyhedra
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Second-order prestress stability and third-order rigidity of polyhedral surfaces
No fractional rigidity orders occur before third order, and second-order prestress stability can be checked by explicit linear-algebra criteria, but a general third-order test exists only when the rigidity matrix has one-dimensional null space.