We study the dynamics of Dehn twists acting on the Dynnikov coordinate plane associated with a thrice-punctured disc. This action, arising from the pure mapping class group, is shown to be piecewise-linear and area-preserving. We describe invariant regions under iteration---termed "black holes"---and present algorithms to compute distances in the curve complex and to track the action of pseudo-Anosov maps. Our results offer new geometric interpretations and reveal structural parallels with toric surfaces and higher-genus covers.