I am taking a course on Topological Graph Theory, where we have looked into the topic of Book Embeddings. The particularly interesting ones were Book Embeddings with thickness 2. This essentially means we embed the graph so that all vertices lie on a straight line $l$, and the edges are arcs (simple curves) on one of the half planes created by this line with no crossings allowed. We know that only graphs who are subgraphs of a hamiltonian planar graph can be drawn this way.
My question is what happens if we allow the edges to cross the line on which the vertices lie. (if possible we also want the edges not to twist so we set a direction on the line $l$ and require the edges to monotonically increase in that direction). We still do not allow the edges to cross each other just the line. Can we then draw all planar graphs this way? Is this question already addressed somewhere?
Also what happens if we want to minimize the crossings on a certain graph. Let's say we only allow the edges to cross the line once and we want to use the least amount of edges that cross the line. Are there bounds on this?
Any embedding can be represented according to your rules.
Represent your embedding as an embedding monotonic in the $x$ direction (Draw your embedding as a Fary embedding for example). Move the vertices a little bit so that no vertices have the same $x$ coordinates. Cut the $x$ axis in $n$ segments corresponding to the projection of the $n$ vertices on the $x$ axis. For each interval $x_i, x_{i+1}$, for each point such that $x_i\leq x < x_{i+1}$, apply the following affine transformation: $$(x, y) \mapsto (x, y - y_i + \frac{x-x_i}{x_{i+1}-x_i}(y_i-y_{i+1})$$ This transformation sends the vertices with abscissa $x_i$ and $x_{i+1}$ to the $x$-axis. It is easy to see that this transformation is continuous and preserves the monotony along the $x$ axis.