Untangling planar curves

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Θ(n^3/2) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n²), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. This lower bound also implies that Ω(n^3/2) degree-1 reductions, series-parallel reductions, and ΔY transformations are required to reduce any planar graph with treewidth Ω(√n) to a single edge, matching known upper bounds for rectangular and cylindrical grid graphs. Finally, we prove that Ω(n²) homotopy moves are required in the worst case to transform one non-contractible closed curve on the torus to another; this lower bound is tight if the curve is homotopic to a simple closed curve.