INTERSECTIONS OF MULTICURVES FROM DYNNIKOV COORDINATES

We present an algorithm for calculating the geometric intersection number of two multicurves on the $n$ -punctured disk, taking as input their Dynnikov coordinates. The algorithm has complexity $O(m^{2}n^{4})$ , where  $m$ is the sum of the absolute values of the Dynnikov coordinates of the two multicurves. The main ingredient is an algorithm due to Cumplido for relaxing a multicurve.