Moves on Curves on Nonorientable Surfaces

Abstract

Let Ng,n denote a nonorientable surface of genus g with n punctures and one boundary component. We give an algorithm to calculate the geometric intersection number of an arbitrary multicurve d. with so-called relaxed curves in Ng,n making use of measured n1-train tracks. The algorithm proceeds by the repeated application of three moves which take as input the measures of d. and produces as output a multicurve d.' which is minimal with respect to each of the relaxed curves. The last step of the algorithm calculates the number of intersections between d.' and the relaxed curves.