Recognising conjugacy classes of Dehn twists on ${\mathbb{D}}_3$

We analyse the action of the basic Dehn twists on the essential curves, $\gamma$, 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 $H_1(\mathrm{T})={\mathbb{Z}}^2$ of the branched covering torus with a hole, $\mathrm{T}\to {\mathbb{D}}_3$. Our explicit description of orbits of the action of the pure mapping class group $\mathrm{P}\mathrm{M}\mathrm{o}\mathrm{d}({\mathbb{D}}_3)$ can be viewed as a solution of the conjugacy problem for the Dehn twists $t_{\gamma }$. We also present an ``untwisting algorithm’’ for factorization of this problem into a minimal number of steps.