Regular projections of links look like graphs in the plane. So I'm wondering if it would be possible to specify a link up to isotopy with purely combinatorial data about this graph. If so, what kind of dictionary is there between graphs of valency 4 and isotopy classes of links? Of course it has to be a one-way dictionary (from graphs to links) since links have many different regular projections. I suppose this may also just be a question about how you know when a graph of valency 4 imbeds in the plane, a question which I assume has some general formulation which has been thoroughly studied. But that would still leave the question of whether isomorphic graphs (with the same "over-under" data for crossings) give isotopic links.
Here is my thinking so far:
If edges are the curves between intersections and nodes are intersections, the associated graph seems to have two kinds of data:
- The adjacency data
- Over-under data at each node (specifying which two incoming edges pass over the other two)
These two sorts of data are easy to translate into graph-theoretic language. The second kind actually has nothing to do with the graph structure and doesn't constrain the graph at all; one can make over-under assignments however one wants and still get a link diagram. But there also seems to be some strange third constraint that captures the "transversality" of the intersections, and I'm not sure how that would be described combinatorially.