Suppose that a foliation $\cal F$ on a surface $F$ is carried by a train track $\tau$. Is it possible to classify all $\cal F$-transverse multi-loops in $F$ in terms of a combinatorial data on $\tau$ (perhaps under some restrictions on $\tau$ or $\cal F$, like $\tau$ being birecurrent, etc)?


For example, every even integral tangential measure $\mu$ on $\tau$ defines a multi-curve $\alpha$ transverse to $\tau$ and, hence, to $\cal F.$ But is such $\mu$ unique for $\alpha$? Also, it seems that not all $\cal F$-transverse curves can be obtained this way in general, since one can obtain a new $\cal F$-transverse multi-curve by twisting $\alpha$ along $\tau$.



Classifying transverse curves to a surface foliation carried by a train track
The train track $\tau$ has a dual bigon track $\tau^\perp$, described in (for example) Penner's book. The bigon track $\tau^\perp$ might not be maximal, but it can be enlarged in various ways by triangulating its complementary regions. Every transverse multiloop to $\tau$ is carried by a unique minimal enlargement of $\tau^\perp$. Furthermore, although the transverse measure that you get on $\tau^\perp$ (or one of its enlargements) is not unique, the bigons can be used to precisely define linear relations, representing the operation of isotoping strands across the bigon. Altogether this gives a parameterization as you like.


