Geometric Intersection Numbers between Multiple and Elementary Curves on Punctured Torus

Abstract

We obtain formulas that give the geometric intersection numbers between multiple curves and elementary curves, which are particular types of multiple curves, on the punctured torus with boundary by using Dynnikov coordinates. To do this, we first determine what the elementary curves on this surface are,
and then find separately the formulas that give the number of geometric intersections of multiple curves with each elementary curve.