# 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.