Enumeration of simple complete topological graphs

A simple topological graph $T=(V(T),E(T))$ is a drawing of a graph in the plane, where every two edges have at most one common point (an end-point or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We prove that the number of isomorphism classes of simple complete topological graphs on n vertices is $2^{\mathrm{\Theta }(n^4)}$. We also show that the number weak isomorphism classes of simple complete topological graphs with n vertices and $\left(\begin{array}{c}\hfill n\hfill \\ \hfill 4\hfill \end{array}\right)$ crossings is at least $2^{n(\log n?O(1))}$, which improves the estimate of Harborth an Mengersen.