Flexible circuits in the d‐dimensional rigidity matroid

Abstract

AbstractA bar‐joint framework in is rigid if the only edge‐length preserving continuous motions of the vertices arise from isometries of . It is known that, when is generic, its rigidity depends only on the underlying graph , and is determined by the rank of the edge set of in the generic ‐dimensional rigidity matroid . Complete combinatorial descriptions of the rank function of this matroid are known when , and imply that all circuits in are generically rigid in when . Determining the rank function of is a long standing open problem when , and the existence of nonrigid circuits in for is a major contributing factor to why this problem is so difficult. We begin a study of nonrigid circuits by characterising the nonrigid circuits in which have at most vertices.