Fáry's theorem says that a simple planar graph can be drawn without crossings so that each edge is a straight line segment.
My question is whether there is an analogous theorem for graphs of bounded crossing number. Specifically, can we say that a simple graph with crossing number k can be drawn so that there are k crossings in the drawing and so that each edge is a curve of degree at most f(k) for some function f?
EDIT: As David Eppstein remarks, it is readily seen that Fáry's theorem implies a drawing of a graph with crossing number k so that each edge is a polygonal chain with at most k bends. I'm still curious though whether each edge can be drawn with bounded degree curves. Hsien-Chih Chang points out that f(k) = 1 if k is 0, 1, 2, 3, and f(k) > 1 otherwise.