Curves on Non-Orientable Surfaces and Crosscap Transpositions

Let N-g,N-n be a non-orientable surface of genus g with n punctures and one boundary component. In this paper, we describe multicurves in N-g,N-n making use of generalized Dynnikov coordinates, and give explicit formulae for the action of crosscap transpositions and their inverses on the set of multicurves in N-g,N-n in terms of generalized Dynnikov coordinates. This provides a way to solve on non-orientable surfaces various dynamical and combinatorial problems that arise in the study of mapping class groups and Thurston's theory of surface homeomorphisms, which were solved only on orientable surfaces before.