Opening cuts for rational mappings

Consider the problem of finding a rational function, whose inverse performs a univalent opening map of the complement of a given set of pairwise disjoint Jordan arcs. The inverse problem is to find the corresponding set of Jordan arcs for an arbitrary rational function with simple critical points. We show how these problems are related to Hurwitz numbers, Dehn twists and other concepts of the theory of Riemann surfaces. The talk is based on joint project with S. Kalmykov, B. Nagy and O. Sete. The research is supported by Moscow Center for Fundamental and Applied Mathematics, agreement with the Ministry of Science and Higher Education of the Russian Federation No. 075-15-2022-283.