Constructing large k-systems on Surfaces

2014/03/20 by Tarik Aougab, Aougab, Tarik
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #math.CO #math.GT

paper · pdf · doi:10.48550/arxiv.1403.5123

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

arxiv created 2016/02/24 · arxiv updated 2016/02/25

Abstract

Let S_g denote the genus g closed orientable surface. For k∈ 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 \citeJu-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≥ 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.

Related

• Algebraic k-systems of curves
• Filling systems of maximum size
• Packing curves on surfaces with few intersections
• Large Collections of Curves Pairwise Intersecting Exactly Once
• Filling systems on surfaces