Detecting Free Products in the Mapping Class Group of Punctured Disks via Dynnikov Coordinates

TL;DR

This paper introduces an algorithm using Dynnikov coordinates to identify free products generated by Dehn twists in the mapping class group of punctured disks, providing a practical tool for understanding their algebraic structure.

Contribution

The paper presents a novel algorithm that determines whether a set of curves forms a complete partition and reveals the structure of the generated free products using Dynnikov coordinates.

Findings

Algorithm accurately verifies complete partitions.

Determines the algebraic structure of Dehn twist groups.

Provides a practical method for analyzing mapping class groups.

Abstract

We prove that Dehn twists about opposite curves that define a complete partition on an $n$-punctured disk $D_n$ generate either a free group or a free product of abelian groups. Additionally, we introduce an algorithm based on Dynnikov coordinates to determine whether a given collection of opposite curves forms a complete partition. This algorithm not only verifies completeness but also reveals the exact structure of the free products generated by these Dehn twists, relying solely on the Dynnikov coordinates of the curves as input.