in this paper we have non-critical level curves.
"On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps at Besançon "
We have :
Let $I=[0,1]$. We consider Morse functions $f:(\Sigma, \partial \Sigma) \rightarrow(I, 0)$ whose critical points all lie in the interior of $\Sigma$. To such a Morse function we associate a finite graph $\Gamma(f)$, which is the quotient space of $\Sigma$ obtained by collapsing all points in the same component of a level set $f^{-1}(a)$ to a single point in $\Gamma(f)$. If we assume $f$ is generic, so that all critical points have distinct critical values, then the vertices of $\Gamma(f)$ all have valence 1 or 3 and arise from critical points of $f$ or from boundary components of $\Sigma$. Namely, boundary components give rise to vertices of valence 1, as do local maxima and minima of $f$, while saddles of $f$ produce vertices of valence 3 . See figure 2 of $[\mathrm{HT}]$ for pictures. We can associate to such a function $f$ a maximal multicurve $C(f)$, unique up to isotopy, by either of the following two equivalent procedures: (1) Choose one point in the interior of each edge of $\Gamma(f)$, take the loops in $\Sigma$ which these points correspond to, then delete those loops which bound disks in $\Sigma$ or are isotopic to boundary components, and replace collections of mutually isotopic loops by a single loop. (2) Let $\Gamma_{0}(f)$ be the unique smallest subgraph of $\Gamma(f)$ which $\Gamma(f)$ deformation retracts to and which contains all the vertices corresponding to boundary components of $\Sigma$. If $\Gamma_{0}(f)$ has vertices of valence 2 , regard these not as vertices but as interior points of edges. In each edge of $\Gamma_{0}(f)$ not having a valence 1 vertex as an endpoint, choose an interior point distinct from the points which were vertices of valence 2 . Then let $C(f)$ consist of the loops in $\Sigma$ corresponding to these chosen points of $\Gamma_{0}(f)$.
Every maximal multicurve arises as $C(f)$ for some generic $f:(\Sigma, \partial \Sigma) \rightarrow(I, 0) .$ To obtain such an $f$, one can first define it near the loops of the given multicurve and the loops of $\partial \Sigma$ so that all these loops are noncritical level curves, then extend to a function defined on all of $\Sigma$, then perturb this function to be a generic Morse function.
what is level curves? what is non-critical level curves?
The classic example of a Morse function is the height on an upright torus. In the diagram, the torus is cut off at the bottom, creating a boundary curve in red; the three critical levels (the two figure-8 curves and the point at top) are blue, typical non-critical levels are green, and the associated graph is at right.