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

2025/01/21 by Elif Medetoğulları, Medetoğulları, Elif, Elif Dalyan +3
Mathematics · #57M50 #57M60 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2501.11912

openalex publication_date 2025/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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