Self-intersecting filling curves on surfaces

Abstract

Let Sg be a closed and oriented surface of genus g ≥ 2. A closed curve γ on Sg is said to fill Sg (or simply be filling), if its complement in the surface is a disjoint union of topological discs. It is assumed that the curve γ 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 ≥ 1, there exists a filling curve γb on Sg 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 Sg minimally.