A Chord Diagrammatic Presentation of the Mapping Class Group of a Once Bordered Surface

TL;DR

This paper presents a new combinatorial presentation of the mapping class group of a once-bordered surface using chord diagrams, enhancing understanding of its structure and applications.

Contribution

It introduces a novel chord diagram-based presentation of the mapping class group, including a dual version with practical advantages.

Findings

Provides a presentation with chord slides as generators

Defines five types of relations for the groupoid

Introduces a dual presentation with specific benefits

Abstract

The Ptolemy groupoid is a combinatorial groupoid generated by elementary moves on marked trivalent fatgraphs with three types of relations. Through the fatgraph decomposition of Teichm\"uller space, the Ptolemy groupoid is a mapping class group equivariant subgroupoid of the fundamental path groupoid of Teichm\"uller space with a discrete set objects. In particular, it leads to an infinite, but combinatorially simple, presentation of the mapping class group of an orientable surface. In this note, we give a presentation of a full mapping class group equivariant subgroupoid of the Ptolemy groupoid of an orientable surface with one boundary component in terms of marked linear chord diagrams, with chord slides as generators and five types of relations. We also introduce a dual version of this presentation which has advantages for certain applications, one of which is given.