generators for the mapping class group of a neighborhood of curves?

Question

Let $S$ be a compact, connected, orientable surface, let $a_1,...,a_k \subset S$ be simple closed curves and let $N = N(a_1 \cup \cdots \cup a_k)$ be a regular neighborhood of $a_1 \cup \cdots \cup a_k$. Is the mapping class group of $N$ (the group of homomorphisms that pointwise fix the boundary up to isotopy pointwise preserving the boundary) generated by Dehn twists along the curves $a_1,...,a_k$?

Answer 1

No, take for example to curves which intersects in one point, $N$ will then be a torus with a boundary component, and a Dehn twist around a curve parallel to the boundary won't be generated by Dehn twists around your curves.

$\mathbf{Edit:}$ As mentioned by the original poster in the comment, the example above does not work. Indeed, the Dehn twist $T_d$ around boundary component of this torus will be given by the chain relation $(T_aT_b)^6=T_d.$ An example of such a mapping class as been described in the comments below.