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Computational Polygonal Entanglement Theory

In this paper we are concerned with motions for untangling polygonal linkages (chains, polygons and trees) in 2 and 3 dimensions. We say that an open, simple polygonal chain can be straightened if it can be continuously reconngured to a sequence of collinear segments in such a way that the rigidity and length of each link and the simplicity of the entire chain are maintained throughout the motion. For a closed chain (simple polygon) untangling means convexiication: reconnguration to a convex polygon. For a tree untangling means \\attening". Linkages that cannot be untangled are called locked. Whether a simple open chain in 2D can be straightened remains a tantalizing open problem. For some special classes of chains it is known that they can be straightened. On the other hand a tree can lock. In 3D both open and closed chains can lock without being knotted. An open chain can be straightened if it has a simple orthogonal projection onto some plane. Furthermore, a planar closed simple chain in 3D can be convexiied in a polynomial number of simple moves. A simple move is one in which only a constant number of joints can rotate at once. In this paper we review these and other recent results. In addition we describe some variants of the problem proposed by Cauchy in 1813 and Erd} os in 1935.