Recognizing whether a measured train track on a surface gives a simple closed curve or a multicurve?

Question

Suppose we are given a measured train track on a (closed, orientable, genus $\geq 2$) surface. I believe there is a canonical way to recover a curve from the traintrack, but is there an algorithm for recognizing whether that curve is a simple curve or a multicurve?

Answer 1

There is such an algorithm, in fact the version of this algorithm on the torus is a generalization of Euclid's Algorithm for determining whether two natural numbers $(m,n)$ are relatively prime: replace the larger by the remainder under division by the smaller, and repeat; or, in simpler terms, subtract the smaller from the larger and repeat.

The general algorithm uses split operations on weighted train tracks, which are described in my article in the Notices of the AMS. The algorithm goes like this.

First do some preprocessing: throw away all branches of the train track with $0$ measure; and then perturb your train track so that all switches have valence 3, incident to two edges in one direction and one edge in the other.

Next, look around your train track for a branch $B$ which is "thick", meaning that at each of the two end switches of $B$, the branch $B$ is on the one-ended side of that switch. See Figure 2 for the picture. Referring to that picture, let the left switch have two incident branches $C,D$ from top to bottom, and let the right switch have two incident branches $E,F$ from top to bottom. Now split the edge $B$: if $C$ has greater measure than $E$, use the right split as shown in the figure; if $E$ has greater measure than $C$, use the left split. If they have equal measure, then you can use a "central split" which is not depicted in that figure, but it can be visualized as just doing either of the left or right splits but then just deleting the new branch.

Now repeat.

The algorithm is guaranteed to terminate, and a simple induction shows that all weighted train tracks in the sequence represent the same multicurve.

Now consider the situation when the algorithm terminates. There are no switches, and so the final weighted train track is a disjoint union of one or more circles. If there are two or more such circles, or if there is a single circle with weight $> 1$, then the multicurve is not simple. Otherwise, if there is a single component with weight $1$, the multicurve is simple.

To see the relation to Euclid's algorithm, simply apply the algorithm described to the torus train track depicted in Figure 1(b).