Multicurves have played a fundamental role in the study of mapping class groups of surfaces since the work of Dehn. A beautiful method of describing such systems on the n-punctured disk is given by the Dynnikov coordinate system. In this talk we describe polynomial time algorithms for calculating the number of connected components of a multi curve, and the geometric intersection number of two multicurves on the n-punctured disk, taking as input their Dynnikov coordinates. This is joint work with Toby Hall.