Let us make two small holes around points $0$ and $1$ on the complex plane and consider non-self-intersecting paths that start on the boundary of one hole and finish at the boundary of the another. It seems,
that all these paths are homotopic to the segment of the real line.


How to prove this? Is there any conceptual explanation of this fact?



Non-self-intersecting paths on $\mathbb{C}\setminus\{0,1\}$
Such a homotopy exists and in fact you can assume that it is an isotopy. This is a "standard fact" in the theory of mapping class groups. See Proposition 2.2 of the "Primer" by Farb and Margalit.


