Hyperbanana Graphs
نویسندگان
چکیده
A bar-and-joint framework is a finite set of points together with specified distances between selected pairs. In rigidity theory we seek to understand when the remaining pairwise distances are also fixed. If there exists a pair of points which move relative to one another while maintaining the given distance constraints, the framework is flexible; otherwise, it is rigid. Counting conditions due to Maxwell give a necessary combinatorial criterion for generic minimal barand-joint rigidity in all dimensions. Laman showed that these conditions are also sufficient for frameworks in R. However, the flexible “double banana” shows that Maxwell’s conditions are not sufficient to guarantee rigidity in R. We present a generalization of the double banana to a family of hyperbananas. In dimensions 3 and higher, these are (infinitesimally) flexible, providing counterexamples to the natural generalization
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