Constructing large k-systems on Surfaces

Abstract

Let $S_g$ denote the genus $g$ closed orientable surface. For $k\in \mathbb{N}$, a $k$-system is a collection of pairwise non-homotopic simple closed curves such that no two intersect more than $k$ times. Juvan-Malnič-Mohar \cite{Ju-Mal-Mo} showed that there exists a $k$-system on $S_g$ whose size is on the order of $g^{k/4}$. For each $k\ge 2$, We construct a $k$-system on $S_g$ with on the order of $g^{\lfloor (k+1)/2\rfloor +1}$ elements. The $k$-systems we construct behave well with respect to subsurface inclusion, analogously to how a pants decomposition contains pants decompositions of lower complexity subsurfaces.

arXiv comment12 pages, 8 figures; revised to acknowledge partial NSF support, fixed typos