Black holes in the Dynnikov Coordinate Plane

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.