Self-intersecting filling curves on surfaces

Let $S_g$ be a closed and oriented surface of genus $g\ge 2$. A closed curve $\gamma$ on $S_g$ is said to fill$S_g$ (or simply be filling), if its complement in the surface is a disjoint union of topological discs. It is assumed that the curve $\gamma$ is always in minimal position. To a filling curve, we associate a number $b$, the number of topological discs in its complement. For $b=1$, such a filling curve is called minimally intersecting. We prove that for every $b\ge 1,$ there exists a filling curve ${\gamma }_b$ on $S_g$ whose complement is a disjoint union of $b$ many topological discs. Furthermore, we provide an upper bound on the number of mapping class group orbits of closed curves which fills $S_g$ minimally.