I would seriously consider the photographic technology suggested by Kevin Buzzard in the comments, if you happen to have a decent digital camera handy (and if you trust your drawing skills). Or, as Peter Shor suggests in the comments, you can use a scanner if you have one handy.† Multiple 2½-dimension diagrams, without tons of regularity, are not fun to draw on a computer --- unless perhaps you have access to touch-screens.
If you want special effects or other benefits of computer-aided graphics badly enough, you can do it directly in LaTeX with TiKZ. I recommend their fairly comprehensive and reasonably readable manual. For sufficiently complicated diagrams, I find that drawing diagrams with TiKZ is relatively painless, and more robust to changes (e.g. in how you want lines to curve, or the positions of nodes if you have them) than ones drawn in xfig.
The over/under relationships can be achieved by drawing the "over" curve-segments twice: once with a thicker white line (or a line matching the background color), and once with a thinner line corresponding to the color of the knot. If you use the PGF extensions (technology which accompanies the TiKZ package), you can quite probably do colorful things such as gradients along the knot curve, if this serves to emphasize features of your knot diagrams.
Good luck with your exam!
†Added after Peter's comment — my workplace at the time of my original post has the odd feature that it is more likely to have convenient cameras than convenient scanners.