Suppose we have a torus $S^1 \times S^1$ (or any other orientable surface) with non empty boundary (meaning, we have holes on the surface). If $a$ and $b$ are a meridian and a parallel respectively then by performing a sequence of Dehn twists along $a$ and $b$ and homeomorphisms isotopic to the identity we can move $a$ on to $b$. I know Dehn twists are supported on an annulus, that means that they will fix the boundary. Is that true for the homeomorphisms that are isotopic to the identity as well. For example, is there a way to slide things on the surface without moving the boundary? I can see it intuitevely but I haven't found a way to rigorously explain it (to myself). Can you please help me.
EDIT. I'll try and be more clear. In the picture it's clear that there is an isotopy that can slide the curve on to a meridian of the Torus. Is there a way to make that isotopic slide while keeping the boundary (black hole) fixed?
