2017/08/23 by Shiv Parsad, Parsad, Shiv, Bidyut Sanki +1
Computer Science · Mathematics · #05C10 #57M15 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1708.06928
openalex publication_date 2017/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Fg be a closed orientable surface of genus g. A set Ω= \ γ1, …, γs\ of pairwise non-homotopic simple closed curves on Fg is called a filling system or simply a filling of Fg, if Fg∖ Ω is a union of b topological discs for some b≥ 1. A filling system is called minimal, if b=1. The size of a filling is defined as the number of its elements. We prove that the maximum size of a filling of Fg with b complementary discs is 2g+b-1. Next, we show that for g≥ 2, b≥ 1 with (g,b)≠ (2,1) (resp. (g,b)=(2,1)) and for each 2≤ s≤ 2g+b-1 (resp. 3≤ s≤ 2g+b-1), there exists a filling of Fg of size s with b complementary discs. Furthermore, we study geometric intersection number of curves in a minimal filling. For g≥ 2, we show that for a minimal filling Ω of size s, the geometric intersection numbers satisfy max \lbrace i(γi, γj)| i≠ j\rbrace≤ 2g-s+1, and for each such s there exists a minimal filling Ω=\lbrace γ1, …, γs \rbrace such that max\lbrace i(γi, γj) | i≠ j\rbrace = 2g-s+1.