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

Abstract

We describe in terms of Dynnikov coordinates the three orbits of the pure mapping class group action on the set of essential curves in a 3-punctured disc ${\mathbb{D}}_3$. For any essential curve $\gamma \subset {\mathbb{D}}_3$ we present an efficient algorithm to untwist $\gamma$ into one of the 3 basic representatives of the orbits. In turn, we give transformation formulas between the Dynnikov and torus ${\mathbb{Z}}^2$-coordinates for multicurves. Besides proving minimality of the algorithm, we give an explicit minimal length formula in terms of the associated even continued fractions.

19 pages, 11 figures. In the last update we moved the "Concluding remarks" into Subsection 3, corrected a misprint and edited an abstract and a few sentences in the introduction