Fig. 5

This figure illustrates the configuration space of a chain. The circles at the extreme left and right indicate the two aligned structures. The original linear interpolation is the black thin line segment between them. This line segment in the configuration space has six self-intersections where it crosses the thin red curves representing self-intersecting configurations. Each Reidemeister move may involve \({\text {MaxLength}}\) residues and can at most deviate a fixed distance, roughly \(1/4{\text {MaxLength}}^2\), from the original linear interpolation. From left to right, the first self-intersection is avoidable by an \(\Omega _1\)-move and the next two by an \(\Omega _2\)-move. If the fourth self-intersection is removed by the short thick light colored \(\Omega _1\)-move, then the fifth cannot be removed. Hence, the \(\Omega _2\)-move is preferred. The last self-intersection involves too many residues to remove, is denoted essential, and the length of the dotted end-contraction avoiding it is calculated