Characterizing homotopy of systems of curves on a compact surface by crossing numbers Original Research Article

Let C1, …, Ck and C′1, …, C′k be closed curves on a compact surface S. We characterize (in terms of counting crossings) when there exists a permutation π of {1, …, k} such that C′π(i) is freely homotopic to Ci or C−1i, for each i = 1, …, k.