Here goes a simplified example, with 2 ellipses.
Draw your 2 ellipses, select them and convert to path (Path->Stroke to path, maybe this is not strictly necessary).
Then get the horizontal (blue) ellipse, duplicate it (Ctrl-D), and offset a couple of times (Ctrl-().
Now, draw a rectangle over the area you want to alter:
Select the offset (thick) ellipse and the rectangle, and intersect both (Path->Intersection).
Now you want to remove this thick section from your red ellipse. Select the red ellipse and then the thick blue section, and get the difference (Path->Difference), to get the following:
Now, you can repeat the procedure for the other crossing.