TITLE: Find all Non-isomorphic good drawings of $K_{3,3}$？
QUESTION [8 upvotes]: Sometimes I look at all non-isomorphic good drawings of graphs on a plane or sphere.
Good drawing means that no edge crosses itself, no two edges cross more than once, and no two edges incident with the same vertex cross each other.
Non-isomorphic means that there is no isomorphism of the drawings as embedded graphs.
My questions are:
How many good drawings are there of $K_{3,3}$?
Is there any website or program that lists all good drawings of graphs with at most $n$ vertices?
The three requirements on a good graph are illustrated below:

For small graphs, there are the following results.
For complete graphs with at most 5 vertices, the list of good drawings is basically clear. They are also summarized in Marcus Sahefer's book on Crossing Numbers of Graphs (CRC Press, 2018):

There are two non-isomorphic good drawings of $K_4$ on the sphere.
There are five non-isomorphic drawings of $K_5$ on the sphere.

Meanwhile $K_{2,3}$ has $4$ good drawings, as described in Michal Stas, "Join Products [K. sub. 2, 3]+[C. sub. n]" (Mathematics v8.6, 2020). They are illustrated in the graphic below:

We can get lists of non-isomorphic graphs from websites (e.g. http://users.cecs.anu.edu.au/~bdm/data/graphs.html) or from software such as maple or geng. How can we get a similar list of non-isomorphic good drawings?

REPLY [5 votes]: The list of nonisomorphic good drawings of $K_{m,n}$ with $2\le m,n \le 3$ appears in the following paper:
Heiko Harborth,
Parity of numbers of crossings for complete n-partite graphs,
Mathematica Slovaca, Vol. 26 (1976), No. 2, 77-95.
Persistent URL: http://dml.cz/dmlcz/136111
The number of nonisomorphic good drawings of $K_{3,3}$ obtained is $102$.
For $K_{2,3}$, there are six nonisomorphic drawings. The two that are missing in Figure 1 referenced in the question may not be realizable with straight-line edges.
The list of nonisomorphic good drawings of graphs with up to five vertices appears in the paper
H-D. O. F. Gronau and H. Harborth,
Numbers of nonisomorphic drawings for small graphs,
Proceedings of the Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989).
Congr. Numer. 71 (1990), 105–114.