---
tag: XH6FFFR8AZ
kind: concept
name: Train track
msc: [57K20, 57M50]
sources:
  - url: https://en.wikipedia.org/wiki/Train_track_(mathematics)
same_as:
  wikidata: Q7832909
  wikipedia: Train track (mathematics)
---

## Summary

A train track on a [[surface|VGJJVBZ1MQ]] is an embedded graph whose edges meet smoothly at the vertices, like railway lines merging at switches.
William Thurston introduced train tracks to describe simple closed curves and measured laminations by finitely many numbers, the weights on the edges.
They are a combinatorial tool in the study of the [[mapping class group|JV5NYG913V]], of pseudo-Anosov maps and of Teichmüller space.

## Explanations

### L1 human

Picture a railway map drawn on a surface.
At each switch, tracks merge tangentially, so a train passing through never makes a sharp turn.
A loop on the surface that runs close to the tracks, following them like a train, is said to be carried by the track.
Recording how many times the loop runs along each piece of track gives a list of numbers that describes the loop.