Complete graphs with 4 and 5 vertices

On the left, you see a graph with four vertices and an edge connecting each vertex, which is called a complete graph on four vertices, [math]K_4[/math]. On the right is a [math]K_5[/math].

Can the vertices be repositioned in either graph so that the edges don't intersect. In graph theory, the edges can be curves, not just straight line segments.