We study the following real version of the famous Abhyankar-Moh Theorem: Which real rational map from the affine line to the affine plane, whose real part is a non-singular real closed embedding of $\mathbb{R}$ into $\mathbb{R}^2$, is equivalent, up to a birational diffeomorphism of the plane, to the linear one? We show that in contrast with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with regular maps where there is only one equivalence class up to isomorphism, there are plenty of non-equivalent smooth rational closed embeddings up to birational diffeomorphisms. Some of these are simply detected by the non-negativity of the real Kodaira dimension of the complement of their images. But we also introduce finer invariants derived from topological properties of suitable fake real planes associated to certain classes of such embeddings.
(Joint Work with Adrien Dubouloz).[-]