My 2010 group will work on problems in topological graph theory, combining knot theory and graph theory. A good place to learn some background is "The Knot Book," by Colin Adams. Check out chapter 8 in particular, dealing with intrinsically linked graphs. We will also discuss issues related to planarity of graphs (see Kuratowski's Theorem in most graph theory books. For graph theory background, I really like Diestel's book, as well as Harary's book.) Non-planarity of graphs and intrinsically linked graphs turn out to be related concepts. We will explore this relation.
Caveat: We will not study knots beyond examining non-trivial links in spatial embeddings of graphs, but we will look at graphs from a very cool topological perspective (in my biased opinion!). You might also want to check out:
Some participants in our 2003 program in the Adirondacks:
Earlier that same day at nearby Owl's Head Mountain: