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Recognising conjugacy classes of Dehn twists on $\mathbb D_3$

Ferihe Atalan, Sergey Finashin

Abstract

We analyse the action of the basic Dehn twists on the essential curves, $γ$, in a disc with 3 marked points, $\mathbb D_3$. In particular, we interpret the induced dynamics on the Dynnikov plane in terms of the standard dynamics in homology $H_1({\rm T})=\mathbb Z^2$ of the branched covering torus with a hole, ${\rm T}\to \mathbb D_3$. Our explicit description of orbits of the action of the pure mapping class group ${\rm PMod}(\mathbb D_3)$ can be viewed as a solution of the conjugacy problem for the Dehn twists $t_γ$. We also present an ``untwisting algorithm'' for factorization of this problem into a minimal number of steps.

Recognising conjugacy classes of Dehn twists on $\mathbb D_3$

Abstract

We analyse the action of the basic Dehn twists on the essential curves, , in a disc with 3 marked points, . In particular, we interpret the induced dynamics on the Dynnikov plane in terms of the standard dynamics in homology of the branched covering torus with a hole, . Our explicit description of orbits of the action of the pure mapping class group can be viewed as a solution of the conjugacy problem for the Dehn twists . We also present an ``untwisting algorithm'' for factorization of this problem into a minimal number of steps.
Paper Structure (33 sections, 25 theorems, 16 equations, 8 figures, 1 table)

This paper contains 33 sections, 25 theorems, 16 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivations and the task
  3. The methods and results
  4. Structure of the paper
  5. Preliminaries
  6. $\mathrm{Mod}(\mathbb D_3)$ and its subgroups
  7. Essential curves and multi-curves
  8. The Dynnikov coordinate system for $\mathbb D_3$.
  9. Geometric intersection numbers after lifting of curves
  10. The update rules.
  11. The Dynnikov Dynamics of the basic Dehn-twists
  12. Geometry of the action of $t_c$ and $t_d$ on the Dynnikov plane
  13. Tracks of orbits of $t_c$ and $t_d$ on the Dynnikov plane
  14. The Dynnikov coordinates on a torus
  15. From torus coordinates to Dynnikov coordinates
  16. ...and 18 more sections

Key Result

Theorem 1.1

Any Dehn twist $t_\gamma\in\mathrm{PMod}(\mathbb D_3)$ about an essential curve $\gamma$ in $\mathbb D_3$ is conjugate to $t_c$, $t_d$, or $t_e$ by an element of $\mathrm{PMod}(\mathbb D_3)$. Namely, in terms of Dynnikov coordinates $(a,b)$ of $\gamma$:

Figures (8)

  • Figure 1: The curves $c$ and $d$ on $\mathbb D_3$
  • Figure 2: The curves $c$ and $d$ on $\mathbb D_3$
  • Figure 3:
  • Figure 4: The linearity regions for the $t_c$ and $t_d$-action
  • Figure 5: Tracks (a) $O^c_n$ of $t_c$-action and (b) $O^d_n$ of $t_d$-action
  • ...and 3 more figures

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Example 2.5
  • Theorem 2.6
  • Lemma 2.7
  • proof
  • ...and 35 more