Configuration spaces of convex and embedded polygons in the plane

A celebrated result of Connelly, Demaine, and Rote [6] states that any polygon in the plane can be “convexified.” That is, the polygon can be deformed in a continuous manner until it becomes convex, all the while preserving the lengths of the sides and without allowing the sides to intersect one another. In the language of topology, their argument shows that the configuration space of embedded polygons with prescribed side lengths deformation retracts onto the configuration space of convex polygons having those side lengths. In particular, both configuration spaces have the same homotopy type. Connelly, Demaine, and Rote observe (without proof) that the space of convex configurations is contractible. Separately, work of Lenhart and Whitesides [10] and of Aichholzer, Demaine, Erickson, Hurtado, Overmars, Soss, and Toussaint [1] had shown that the space of convex configurations is connected. These results are part of the literature on linkages. The polygons here are mechanical linkages in which the sides can be viewed as rigid bars of fixed length arranged in a cycle and the vertices as joints about which the bars can rotate. In this note we determine the topology of the space of convex configurations and the space of embedded configurations up to homeomorphism. We regard two polygons as equivalent if one can be translated and rotated onto the other. After a translation, we may assume that one of the vertices is at the origin and then, after a rotation, that one of the adjacent sides lies along the positive x-axis. To fix some notation, let ~̀ = (`1, . . . , `n) be a given sequence of side lengths, where `i > 0 for all i and n ≥ 3. The configuration space of planar polygons with these side lengths is defined by: