  Topological drawings are natural representations of graphs in the plane,
where vertices are represented by points, and edges by curves connecting the
points. Topological drawings of complete graphs and of complete bipartite
graphs have been studied extensively in the context of crossing number
problems. We consider a natural class of simple topological drawings of
complete bipartite graphs, in which we require that one side of the vertex set
bipartition lies on the outer boundary of the drawing.
  We investigate the combinatorics of such drawings. For this purpose, we
define combinatorial encodings of the drawings by enumerating the distinct
drawings of subgraphs isomorphic to $K_{2,2}$ and $K_{3,2}$, and investigate
the constraints they must satisfy. We prove that a drawing of $K_{k,n}$ exists
if and only if some simple local conditions are satisfied by the encodings.
This directly yields a polynomial-time algorithm for deciding the existence of
such a drawing given the encoding. We show the encoding is equivalent to
specifying which pairs of edges cross, yielding a similar polynomial-time
algorithm for the realizability of abstract topological graphs.
  We also completely characterize and enumerate such drawings of $K_{k,n}$ in
which the order of the edges around each vertex is the same for vertices on the
same side of the bipartition. Finally, we investigate drawings of $K_{k,n}$
using straight lines and pseudolines, and consider the complexity of the
corresponding realizability problems.
