Locked and Unlocked Polygonal Chains in 3D
Résumé
In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be {\em straightened\/} if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is {\em convexification\/}: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called {\em locked}. While there are open chains in 3D that are locked, we show that if an open chain has a simple orthogonal projection onto some plane, it can be straightened. For closed chains, we show that there are unknotted but locked closed chains, and we provide an algorithm for convexifying a planar simple polygon in 3D with a polynomial number of moves.