---
tag: P8055074WH
kind: definition
name: Train track
msc: [57K20, 57M50]
defines: XH6FFFR8AZ
uses: [VGJJVBZ1MQ, 00C0BR7D7H]
sources:
  - url: https://en.wikipedia.org/wiki/Train_track_(mathematics)
---

## Definition

Let $S$ be a smooth surface.
A *train track* in $S$ is a finite [[graph|00C0BR7D7H]] $\tau$ embedded in $S$, with vertices called *switches* and edges called *branches*, such that:

1. each branch is a smoothly embedded arc, and at each switch all branches that end there are tangent to one common line;
2. at each switch, at least one branch arrives from each of the two directions along this line;
3. no component of $S \setminus \tau$ is an open disc with $0$, $1$ or $2$ cusps, or an open annulus with no cusps.

Here a *cusp* of a complementary region is a corner of its boundary at a switch where two branches arriving from the same direction meet.

## Remarks

Clauses 1 and 2 let a smooth path run along $\tau$ through each switch without turning back.
Clause 3 rules out regions that would let curves carried by $\tau$ be shortened or be parallel copies of each other.
A *weight* on $\tau$ assigns a number $w(b) \ge 0$ to each branch $b$.
At each switch, the total weight arriving from one side must equal the total arriving from the other side.
Integer weights correspond to multicurves carried by $\tau$.

Example: let $S$ be a closed orientable surface of genus $2$ and $c$ a non-separating simple closed curve.
Put one switch on $c$; the rest of $c$ is one branch leaving the switch in one direction and coming back from the other, so clauses 1 and 2 hold.
The complement $S \setminus c$ is a surface of genus $1$ with two boundary circles, neither a disc nor an annulus, so clause 3 holds.

Non-example: two disjoint parallel copies of $c$, each with one switch, bound an annulus with no cusps between them, so clause 3 fails.

## Explanations

### L1 human

A train track is a smooth railway network on a surface.
The rules say that trains can pass through every junction smoothly, and that the network has no useless loops or parallel duplicate lines.
Numbers on the tracks, balanced at each switch, describe how a family of curves runs along the network.