On topological and geometric (194)(194) configurations

An (nk)(nk) configuration is a set of  nn points and  nn lines such that each point lies on  kk lines while each line contains  kk points. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of (nk)(nk) configurations for a given  kk has been subject to active research. A current front of research concerns geometric (n4)(n4) configurations: it is now known that geometric (n4)(n4) configurations exist for all  n≥18n≥18, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of (194)(194) configurations: we obtain all topological (194)(194) configurations among which none are geometrically realizable.