Hyperbanana Graphs
Abstract
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 bar-and-joint rigidity in all dimensions. Laman showed that these conditions are also sufficient for frameworks in R^2. However, the flexible "double banana" shows that Maxwell's conditions are not sufficient to guarantee rigidity in R^3. 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 of Laman's theorem.
Keywords
Cite
@article{arxiv.1308.3281,
title = {Hyperbanana Graphs},
author = {Christopher Clement and Audrey Lee-St. John and Jessica Sidman},
journal= {arXiv preprint arXiv:1308.3281},
year = {2013}
}