The expected genus of a random chord diagram

TL;DR

This paper investigates the average genus of random chord diagrams, establishing that it grows approximately as half the number of chords minus a logarithmic correction, revealing asymptotic behavior.

Contribution

It provides an asymptotic formula for the expected genus of random chord diagrams, connecting combinatorial structures with topological properties.

Findings

Expected genus g_n ≈ n/2 - Theta(ln n)

Asymptotic behavior of genus for large n

Link between chord diagrams and surface topology

Abstract

To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g_n denote the expected genus of a randomly chosen oriented chord diagram of order n. We show that g_n satisfies: g_n = n/2 - Theta(ln n).