The Expected Genus of a Random Chord Diagram

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 gn denote the expected genus of a randomly chosen oriented chord diagram of order n. We show that gn satisfies: gn = n 2 − Θ(ln n). I.e., there exist 0 < c1 < c2 and n0 such that c1 ln n ≤ n 2 − gn ≤ c2 ln n for all n ≥ n0.

• Type: Journal
• Year: 2011
• Where: DCG
• Authors: Nathan Linial, Tahl Nowik
• Keywords: Computer Graphics, DCG 2011, Generic Curve, Oriented Chord Diagram, Oriented Surface