2026/03/15 by Ferihe Atalan, Sergey Finashin
Mathematics · #math.GT
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 D3. For any essential curve γ⊂\mathbb D3 we present an efficient algorithm to untwist γ into one of the 3 basic representatives of the orbits. In turn, we give transformation formulas between the Dynnikov and torus \mathbb Z2-coordinates for multicurves. Besides proving minimality of the algorithm, we give an explicit minimal length formula in terms of the associated even continued fractions.