Genus Ranges of Chord Diagrams

TL;DR

This paper introduces the concept of genus ranges of chord diagrams, exploring their possible values through theoretical analysis and computer-aided calculations, revealing which intervals can be realized as genus ranges.

Contribution

It defines and investigates the genus range of chord diagrams, including characterizing which integer intervals can be realized, supported by computational methods.

Findings

Identified which integer intervals can be realized as genus ranges.

Developed computational tools to analyze genus ranges.

Provided theoretical and computational insights into the properties of genus ranges.

Abstract

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can, and cannot, be realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.