Before revision:  Finally, we show that some graphs have a unique fan-crossing free embedding, that there are  maximal fan-crossing free graphs with less edges than the density , and that the recognition problem for fan-crossing free graphs is NP-complete.
After revision:  Finally, we show that some graphs have a unique fan-crossing free embedding, that there are thinned maximal fan-crossing free graphs with less edges than the density , and that the recognition problem for fan-crossing free graphs is NP-complete.