The Expected Genus of a Random Chord Diagram

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: $$g_n=\frac{n}{2}-\varTheta (\ln n).$$ I.e., there exist 0<c 1<c 2 and n 0 such that \(c_{1}\ln n\leq\frac{n}{2}-g_{n}\leq c_{2}\ln n\) for all n≥n 0.