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

Elif Medetoğulları, Elif Dalyan, S. Öykü Yurttaş

TL;DR

This work analyzes subgroups of the mapping class group of an $n$-punctured disk generated by Dehn twists along opposite curves. It shows that, whenever the curves admit a complete partition, the generated subgroup decomposes as a free product of abelian factors $\mathbb{Z}^{n_i}$, and becomes a free group of rank $k$ when the family is maximal. A central contribution is an algorithm that uses Dynnikov coordinates to test completeness of the partition and to extract the precise free-product (or free) structure from the input curves alone. The approach hinges on a Dynnikov-coordinate–driven Ping-Pong framework, introducing decisive sets and $X_i$-domains to certify freeness, and provides concrete steps to detect loop types and oppositeness computationally. This yields both structural classification and a practical procedure for identifying free (products) subgroups in $\mathrm{Mod}(D_n)$ generated by Dehn twists.

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.

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

TL;DR

This work analyzes subgroups of the mapping class group of an -punctured disk generated by Dehn twists along opposite curves. It shows that, whenever the curves admit a complete partition, the generated subgroup decomposes as a free product of abelian factors , and becomes a free group of rank when the family is maximal. A central contribution is an algorithm that uses Dynnikov coordinates to test completeness of the partition and to extract the precise free-product (or free) structure from the input curves alone. The approach hinges on a Dynnikov-coordinate–driven Ping-Pong framework, introducing decisive sets and -domains to certify freeness, and provides concrete steps to detect loop types and oppositeness computationally. This yields both structural classification and a practical procedure for identifying free (products) subgroups in generated by Dehn twists.

Abstract

We prove that Dehn twists about opposite curves that define a complete partition on an -punctured disk 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.
Paper Structure (11 sections, 4 theorems, 18 equations, 8 figures)

This paper contains 11 sections, 4 theorems, 18 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Dynnikov coordinate system
  4. Family of opposite curves
  5. Tools for the Main Theorem
  6. Complete partition and decisive sets
  7. Main Results
  8. Algorithm
  9. Detecting right and left loops
  10. Test for opposite curves
  11. Statement of the algorithm

Key Result

Theorem 1.1

Let $\mathcal{C}=\{c_1, c_2, \ldots c_k\}$ be a family of opposite curves on $D_n$. Suppose that $\mathcal{P}=\{P_1, P_2, \ldots , P_m \}$ is a complete partition of $\mathcal{C}$ where $|P_i|=n_i$$(1\leq i\leq m)$. Then $\langle t_{c_1}, \ldots, t_{c_k} \rangle$ is isomorphic to the free product $\

Figures (8)

  • Figure 1: The arcs $\alpha_i$ and $\beta_i$
  • Figure 2: Large left loop, $L_{i,j}(c)$ and large right loop, $R_{i,j}(c)$ of $c \cap\Delta_{i,j}$
  • Figure 3: Opposite large loops, $R_{i,j}(c_1)$ and $L_{i,j}(c_2)$ in $\Delta_{i,j}$
  • Figure 4: Maximal family of opposite curves
  • Figure 5: Curve surgery
  • ...and 3 more figures

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 11 more