Black holes in the Dynnikov Coordinate Plane

TL;DR

This work analyzes the dynamics of Dehn twists on the Dynnikov coordinate plane for a thrice-punctured disc $M$, showing that the action of ${\rm PMod}(M)$ via $t_c$ and $t_d$ is a piecewise-linear, area-preserving ${\mathbb Z}^2$-automorphism with a fixed fan structure. It provides explicit update rules for $t_c^2$ and $t_d^2$, describes invariant quadrants (black holes), and derives detailed orbits of the generators on $\mathbb{Z}^2$, including compositions such as $t_c t_d$. The paper then connects these dynamics to the curve complex $ extbf{C}(M)$ by encoding vertices with Dynnikov coordinates and offers an algorithm to compute distances to the distinguished triple $\Delta=ig\{v_c,v_d,v_e\big\}$. It proves that the subgroup generated by $t_c^2$, $t_d^2$, and $t_e^2$ is free of rank $3$ using the Ping-Pong Lemma, and presents procedures to study pseudo-Anosov maps like $t_d t_c^{-1}$ and $t_d^{-1} t_c$ via orbit analysis, including a Diophantine pattern in their coordinate sequences. An appendix relates Dynnikov coordinates to $(p,q)$-coordinates on the one-holed torus through a double branched cover, linking the plane calculus to a classical surface model.

Abstract

This work presents an application of Dynnikov coordinates in geometric group theory. We describe the orbits and dynamics of the action of Dehn twists $t_c$ and $t_d$ in the Dynnikov coordinate plane for a thrice-punctured disc $M$, where $c$ and $d$ are simple closed curves with Dynnikov coordinates $(0,1)$ and $(0,-1)$, respectively. This action has an interesting geometric meaning as a piecewise linear $\mathbb{Z}^{2}$-automorphism preserving the shape of the linearity border fan.

Figures (13)

• Figure 1: Two curves $c$ and $d$ on $M$
• Figure 2:
• Figure 3: Two curves $c$ and $d$ on $M$
• Figure 4:
• Figure 5:

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• proof
• Lemma 5.1
• Remark 6.1
• Remark 6.2