Untangling planar curves (Q1688858)

A homotopy move is a local transformation taking a closed planar curve to a simple closed curve. The authors prove that to simplify a planar closed curve having \(n\) self crossings needs at most \(O(n^{3/2})\) homotopy moves. Hence they improve the previous bound of \(O(n^2)\). They also improve some other bounds similarly like the bound for a transformation taking one immersion of \(k\) circles having at most \(n\) self crossings into another. Also they prove that transforming a non-contractible closed curve to another one on an orientable surface needs a maximum of \(\Omega(n^2)\) homotopy moves. This is a nice paper combining surface topology, graph theory and combinatorics.