We present a study on how the outcomes of binary-single interactions involving three black holes (BHs) distribute as a function of the initial conditions; a distribution we refer to as the topology. Using a N-body code that includes BH finite sizes and gravitational wave (GW) emission in the equation of motion (EOM), we perform more than a million binary-single interactions to explore the topology of both the Newtonian limit and the limit at which general relativistic (GR) effects start to become important. From these interactions, we are able to describe exactly under which conditions BH collisions and eccentric GW capture mergers form, as well as how GR in general modifies the Newtonian topology. This study is performed on both large- and microtopological scales. We further describe how the inclusion of GW emission in the EOM naturally leads to scenarios where the binary-single system undergoes two successive GW mergers.