Genus Ranges of Chord Diagrams

Area of Science:

• Topology
• Combinatorics
• Computational Geometry

Background:

• Chord diagrams are topological objects defined by a circle (backbone) and chords connecting points on the circle.
• The genus of a chord diagram is determined by the orientable surface formed by thickening the backbone and attaching bands representing chords.
• Variations exist where bands attach to either the inner or outer boundary of the thickened backbone.

Purpose of the Study:

• To investigate the genus ranges of chord diagrams for a fixed number of chords.
• To identify which integer intervals can be realized as genus ranges.
• To explore the properties of genus ranges through computational analysis.

Main Methods:

• Definition of chord diagram genus based on surface topology.
• Exploration of variations in band attachment (inner vs. outer boundary).
• Systematic study of genus ranges for a fixed number of chords.
• Utilizing computer calculations to discover and prove properties.

Main Results:

• Characterization of genus ranges for chord diagrams.
• Identification of realizable and unrealizable integer intervals as genus ranges.
• Demonstration of the utility of computational methods in topological studies.

Conclusions:

• The genus range of a chord diagram is a fundamental property influenced by chord configurations and band attachment variations.
• Computational approaches are crucial for understanding the complex combinatorial and topological properties of chord diagrams.
• This research contributes to the understanding of topological invariants and their realizability in geometric structures.