Black holes in the Dynnikov Coordinate Plane

TL;DR

This paper explores the dynamics of Dehn twists in the Dynnikov coordinate plane for a thrice-punctured disc, revealing their geometric interpretation as piecewise linear automorphisms that preserve certain structures.

Contribution

It introduces a novel application of Dynnikov coordinates to analyze Dehn twist actions and their geometric properties in a specific topological setting.

Findings

Dehn twists act as piecewise linear automorphisms in Dynnikov coordinates.

The action preserves the shape of the linearity border fan.

Provides a geometric interpretation of the dynamics of these twists.

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.