Filling systems on surfaces

Abstract

Let [Formula: see text] be a closed orientable surface of genus [Formula: see text]. A set [Formula: see text] of pairwise non-homotopic simple closed curves on [Formula: see text] is called a filling system or simply be filling of [Formula: see text], if [Formula: see text] is a disjoint union of [Formula: see text] topological discs for some [Formula: see text]. A filling system is called minimally intersecting, if the total number of intersection points of the curves is minimum, or equivalently [Formula: see text]. The size of a filling system is defined as the number of its elements. We prove that the maximum possible size of a minimally intersecting filling system is [Formula: see text]. Next, we show that for [Formula: see text] with [Formula: see text], there exists a minimally intersecting filling system on [Formula: see text] of size [Formula: see text]. Furthermore, we study geometric intersection number of curves in a minimally intersecting filling system. For [Formula: see text], we show that for a minimally intersecting filling system [Formula: see text] of size [Formula: see text], the geometric intersection numbers satisfy [Formula: see text], and for each such [Formula: see text] there exists a minimally intersecting filling system [Formula: see text] such that [Formula: see text].