A combinatorial representation of curves using train tracks (Q1579823)

The author studies curves that are quasi-transversal to a given train track on a closed surface \(M\) of genus \(g\geq 2\). He shows that every homotopy class of a curve on \(M\) is represented by a quasi-transversal curve and that this representation is unique after a normalization. As an application he gives combinatorial algorithms to decide whether a given homotopy class can be represented by a simple closed curve and whether a given mapping class is periodic, pseudo-Anosov, or reducible.