I think knot theory has been studied for quite a while (like a century or so), so I'm just wondering whether there is a "knot theory" for graphs, i.e. the study of (topological properties of) embeddings of graphs into R^3 or S^3.
If yes, can anyone show me any reference?
If the answer is basically no, then why? Is it just too hard, uninteresting, or can it be essentially reduced to the study of knots (and links)?
Best Answer
The theory of knotted trees is obviously trivial. So given a knotted graph $\Gamma$, take a maximal tree in it and you can bring it to a standard form, say to be embedded as a planar object inside a tiny disk that is disjoint from the rest of the knotted graph; which is just the finitely many arcs that make the complement of the tree. But now you can draw $\Gamma$ in the plane so that "everything interesting" (namely, the complement of the tree) is outside of a small disk. Do inversion, and you have a fixed tree outside the disk and a tangle inside it. (Some details depend on whether your vertices are rigid or not, or "thickened" or not, but the conclusion is always more or less the same).
This correspondence between knotted graphs and tangles is not canonical - it depends on the (combinatorial) choice of a maximal tree, and modifying that choice modifies the resulting tangle (in simple ways that will not be stated here).
So topologically speaking, "knotted graphs" are not interesting. They are merely tangles, along with a bit of further combinatorial data (mostly the tree). If you totally understand the theory of tangles (modulo some simple to state actions, which also depend on what rigidity assumptions are made for the vertices), you'd totally understand knotted graphs.
Yet there's lot's of beautiful information in the interaction between the combinatorics of the graph and the topology of the tangle. For example, see my recent paper with Zsuzsanna Dancso, arXiv:1103.1896, in which we study the relationship between knotted trivalent graphs and Drinfel'd associators.