Train track (mathematics)

In the mathematical area of topology, a train track is a family of curves embedded on a surface, meeting the following conditions: The main application of train tracks in mathematics is to study laminations of surfaces, that is, partitions of closed subsets of surfaces into unions of smooth curves.

The study of train tracks was originally motivated by the following observation: If a generic lamination on a surface is looked at from a distance by a myopic person, it will look like a train track.

For instance, Penner and Harer require that each such component, when glued to a copy of itself along its boundary to form a smooth surface with cusps, have negative cusped Euler characteristic.

A train track is said to carry a lamination if there is a train track neighborhood such that every leaf of the lamination is contained in the neighborhood and intersects each vertical fiber transversely.

If each vertical fiber has nontrivial intersection with some leaf, then the lamination is fully carried by the train track.