at.algebraic topology – Untangling two simple closed curves on a surface


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 $#(gamma_1cap gamma_2)>#(gamma_1’cap gamma_2)$.

Question. Is it true that there is a disk on $Ssetminus (gamma_1cupgamma_2)$ 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.