Flexible circuits in the d-dimensional rigidity matroid

Abstract

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