Let $S$ be a smooth surface and $\gamma\_1, \gamma\_2$ be two transversal simple closed curves on it. Suppose moreover that there exists a simple closed curve $\gamma\_1'$ on $S$ isotopic to $\gamma\_1$ and such that $\#(\gamma\_1\cap \gamma\_2)>\#(\gamma\_1'\cap \gamma\_2)$.


**Question.** Is it true that there is a disk on $S\setminus (\gamma\_1\cup\gamma\_2)$ whose boundary is composed of one arc of $\gamma\_1$ and one arc of $\gamma\_2$?


Note that in case such a disk exists, one can construct an isotopy of $\gamma\_1$ that would decrease the number of intersections of $\gamma\_1$ with $\gamma\_2$ by two.



Untangling two simple closed curves on a surface
This is also proved as Lemma 3.1 in
Joel Hass and Peter Scott,
Intersections of curves on surfaces, Israel Journal of Mathematics 51 (1985), 90–120. <https://doi.org/10.1007/BF02772960>


