of oriented intervals $\{I_i\}_{i=1}^{l},$ where $l,$ which may be infinity, is the number of components in the complement of $C$ on $\alpha_0.$ Henceforth, we assume that $2\leq l\leq \infty.$ Label the endpoints of $I_i,\, e_i^1 $ and $e_i^2,$ where $I_i$ is oriented from $e_i^1 $ to $e_i^2.$ Through each point $e_i^j$ there is a unique biinfinite geodesic $\epsilon_i^j$ orthogonal to $\alpha_0$ and oriented in the direction of the end $A^+$, with $\epsilon_i^j(0)=e_i^j.$ Except in the case where two geodesics with different names coincide, these geodesics are pairwise disjoint and the two ends of each $\epsilon_i^j$ go out the two ends of the cylinder $A$. We shall refer to the the union of the geodesics $\epsilon_i^j$ and the intervals $I_i$ as the [*scaffolding*]{}. Note that the scaffolding is completely determined by the choice of the point $p,$ the orientation on $\alpha_0$ the length $|\alpha_0|$ and the set $C$.

Let $E_i$ denote the hyperbolic strip in $A$ bounded by the geodesics $\epsilon_i^1$ and $\epsilon_i^2$ and containing the interval $I_i.$ The scaffolding structure will serve as a foundation for the construction in which the hyperbolic strip $E_i$ shall be removed and then replaced by an untwisted flute surface which has been sliced open 