A combinatorial representation of curves using train tracks

Let M be a closed surface of genus ≥2 . For a train track τ on M , we give a kind of standardization of curves on M with respect to τ . Standardized curves are called quasi-transverse curves, which are analogues to geodesics. Such a curve exists for every, not only simple, homotopy class of curves and is almost unique. If M is orientable in addition, we can further normalize such a curve and obtain a unique representation. are included applications of this normalization to geometric intersections of curves and to carrying of train tracks. The former is an algorithm which examines whether a given element of π1(M) is representable by a simple closed curve, or similar problems. The latter is an algorithm which examines whether a train track is carried by another train track. This leads to an algorithm for recognizing the Thurston type of a given mapping class, where Thurston type is either periodic, pseudo-Anosov or reducible. All algorithms are combinatorial.