Topological fluid mechanics in the sense of the present paper is the study and classification of flow patterns close to a critical point. Here we discuss the topology of steady viscous incompressible axisymmetric flows in the vicinity of the axis. Following previous studies the velocity field $v$ is expanded in a Taylor series at a point on the axis, and the expansion coefficients are considered as bifurcation parameters. After a normal form transformation we easily obtain the most common bifurcations of the flow patterns. The use of non-linear normal forms provide a gross simplification, which to the authors knowledge has not been used systematically to high orders in topological fluid mechanics. We compare the general results with experimental and computational results on the Vogel-Ronneberg flow. We show that the topology changes observed when recirculating bubbles on the vortex axis are created and interact follow the topological classification and that the complete set of patterns found is contained in a codimension-4 unfolding of the most simple singular configuration.