  Let $G$ be a simple topological graph and let $\Gamma$ be a polyline drawing
of $G$. We say that $\Gamma$ \emph{partially preserves the topology} of $G$ if
it has the same external boundary, the same rotation system, and the same set
of crossings as $G$. Drawing $\Gamma$ fully preserves the topology of $G$ if
the planarization of $G$ and the planarization of $\Gamma$ have the same planar
embedding. We show that if the set of crossing-free edges of $G$ forms a
connected spanning subgraph, then $G$ admits a polyline drawing that partially
preserves its topology and that has curve complexity at most three (i.e., at
most three bends per edge). If, however, the set of crossing-free edges of $G$
is not a connected spanning subgraph, the curve complexity may be
$\Omega(\sqrt{n})$. Concerning drawings that fully preserve the topology, we
show that if $G$ has skewness $k$, it admits one such drawing with curve
complexity at most $2k$; for skewness-1 graphs, the curve complexity can be
reduced to one, which is a tight bound. We also consider optimal $2$-plane
graphs and discuss trade-offs between curve complexity and crossing angle
resolution of drawings that fully preserve the topology.
