Around a conjecture by R. Connelly, E. Demaine, and G. Rote

Abstract

Denote by $M(P)$ the configuration space of a planar polygonal linkage, that is, the space of all possible planar configurations modulo congruences, including configurations with self-intersections. A particular interest attracts its subset $M^o(P) \subset M(P)$ of all configurations \emph{without} self-intersections. R. Connelly, E. Demaine, and G. Rote proved that $M^o(P)$ is contractible and conjectured that so is its closure $\bar{M^o(P)}$. We disprove this conjecture by showing that a special choice of $P$ makes the homologies $H_k(\bar{M^o(P)})$ non-trivial.