  An $(n_k)$ configuration is a set of $n$ points and $n$ lines such that each
point lies on $k$ lines while each line contains $k$ 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 $(n_k)$ configurations for a given $k$ has been
subject to active research. A current front of research concerns geometric
$(n_4)$ configurations: it is now known that geometric $(n_4)$ configurations
exist for all $n \ge 18$, apart from sporadic exceptional cases. In this paper,
we settle by computational techniques the first open case of $(19_4)$
configurations: we obtain all topological $(19_4)$ configurations among which
none are geometrically realizable.
