I have a surface $M$ (without boundary) and a homeomorphism $f:M \rightarrow M$, which is isotopic to the identity on $M$. If I delete two points $x$ and $f(x)$ from the surface, I get a homeomorphism $$\tilde{f} : M \backslash\{x,f(x)\} \rightarrow M \backslash\{x,f(x)\}$$
by restricting the homeomorphism $f$ to the punctured surface. My question is now:
Is $\tilde{f}$ also isotopic to the identity on $M \backslash\{x,f(x)\}$?
Could it be possible to perturb the original isotopy to get the desired new one?
Many thanks!