Train track

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.