Three different homotopy classes of paths connecting two regions along a cylindrical surface (schematically representing an activated cluster).

In general, the homotopy class of a path connecting two points on a cylinder is determined by the winding number of the path, i.e., the number of times the path winds around the central axis of the cylinder. This number is an integer: one can count clockwise winding as positive and counterclockwise as negative, or vice versa. The choice of orientation (clockwise vs. anti-clockwise) gives the two isomorphisms of the fundamental group of a cylindrical surface with the integers under addition.