Flexible circuits in the d-dimensional rigidity matroid

A bar-joint framework ( G , p ) in R d is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of R d. It is known that, when ( G , p ) is generic, its rigidity depends only on the underlying graph G, and is determined by the rank of the edge set of G in the generic d-dimensional rigidity matroid Script capital R d. Complete combinatorial descriptions of the rank function of this matroid are known when d = 1 , 2, and imply that all circuits in Script capital R d are generically rigid in R d when d = 1 , 2. Determining the rank function of Script capital R d is a long standing open problem when d >= 3, and the existence of nonrigid circuits in Script capital R d for d >= 3 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 Script capital R d which have at most d + 6 vertices.