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

TL;DR

This paper investigates the topological properties of the configuration space of planar polygonal linkages, disproving a conjecture that its closure is contractible by showing it has non-trivial homology.

Contribution

It provides a counterexample to the conjecture that the closure of the non-self-intersecting configuration space is contractible, revealing non-trivial homology.

Findings

Disproved the conjecture about contractibility of the closure of $M^o(P)$

Identified a specific polygonal linkage with non-trivial homology in its configuration space

Showed that the homology groups $H_k(ar{M^o(P)})$ are non-trivial for some $k$

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.