A combinatorial algorithm for immersed loops in surfaces

In this paper, we develop a purely combinatorial algorithm which minimizes the number of double points of an immersed loop in a closed, orientable surface and converts between the ambient isotopy classes of two homotopic loops using an explicit sequence of elementary homotopies. We note that by introducing a curve-shortening flow known as the disc flow, Hass and Scott have shown that given a pair of general position, immersed loops, each with k double points, then they are homotopic through loops with at most k double points, and that this homotopy may be assumed to be regular except at finitely many points. We demonstrate this here without recourse to the geometry of the surface, by giving an explicit homotopy, which relies solely on the notion of a spanning disc for an immersed loop, which we define here to be an embedding of the standard 2-disc into the universal covering space of the surface, for which the boundary is mapped into the union of the lifts of the loop. The most important of these spanning discs have a 1-gon, 2-gon or 3-gon structure relative to the family of covering curves for the loop.