  Topology is a promising approach toward to the light ring in a generic black
hole background, and equatorial timelike circular orbit in a stationary black
hole background. In this paper, we consider the distinct topological
configurations of the timelike circular orbits in static, spherically
symmetric, and asymptotic flat black holes. By making use of the equation of
motion of the massive particles, we construct a vector with its zero points
exactly relating with the timelike circular orbits. Since each zero point of
the vector can be endowed with a winding number, the topology of the timelike
circular orbits is well established. Stable and unstable timelike circular
orbits respectively have winding number +1 and -1. In particular, for given
angular momentum, the topological number of the timelike circular orbits also
vanishes whether they are rotating or not. Moreover, we apply the study to the
Schwarzschild, scalarized Einstein-Maxwell, and dyonic black holes, which have
three distinct topological configurations, representations of the radius and
angular momentum relationship, with one or two pairs timelike circular orbits
at most. It is shown that although the existence of scalar hair and
quasi-topological term leads to richer topological configurations of the
timelike circular orbits, they have no influence on the total topological
number. These results indicate that the topological approach indeed provides us
a novel way to understand the timelike circular orbits. Significantly,
different topological configurations can share the same topology number, and
hence belong to the same topological class. More information is expected to be
disclosed when other different topological configurations are present.
