We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an $n$-times punctured disk. This provides a way to find the geometric intersection number of two arbitrary integral laminations when combined with an algorithm of Dynnikov and Wiest.
"Geometric intersection of curves on punctured disks." J. Math. Soc. Japan 65 (4) 1153 - 1168, October, 2013. https://doi.org/10.2969/jmsj/06541153