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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 on $\tau^\perp$ that you get 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.