Algebraic k-systems of curves

2019/11/19 by Daly, Charles, Gaster, Jonah, Lahn, Max +2
#Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1911.08469

Abstract

A collection Δ of simple closed curves on an orientable surface is an algebraic k -system if the algebraic intersection number ⟨ α,β⟩ is equal to k in absolute value for every α, β∈ Δ distinct. Generalizing a theorem of [MRT14] we compute that the maximum size of an algebraic k-system of curves on a surface of genus g is 2g+1 when g≥ 3 or k is odd, and 2g otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as g².

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