Methods for counting the intersections of slopes in the flat torus

We define slopes in the flat torus as the set of equivalence classes of the solutions of linear equations in \mathbb{R}^۲. The definition is equivalent to that of closed geodesics in the flat torus passing through the equivalence class of the point (۰,۰). In this paper we derive formulas for counting the number of points in the intersection of multiple slopes in the flat torus.

Keywords:

Torus , intersection of curves , counting methods