Intersections of multicurves from Dynnikov coordinates

Abstract

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 $O(m^2n^4)$, 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.

9 pages. Corrected error in paragraph 1 about complexity of [2] and [9], with thanks to Mark Bell and Saul Schleimer