A combinatorial algorithm for immersed loops in surfaces
Starting with an arbitrary general position immersion of an oriented loop into a closed orientable surface, the author presents a combinatorial algorithm which homotops the loop to one with a minimal number of double points. The homotopy uses elementary Reidemeister-like moves. This approach is generalized to develop an algorithm which, given two homotopic immersed loops, homotops the first loop through a sequence of these elementary moves to produce a loop which is ambient isomorphic to the second. If both loops have \(k\) self-intersections, then the number of self-intersections remains equal to \(k\) throughout the conversion sequence. This gives an explicit homotopy that realizes a result of \textit{J. Hass} and \textit{P. Scott} [Topology 33, 25-43 (1994; Zbl 0798.58019)].
scientific article; zbMATH DE number 1846687