Knots and graphs Every knot gives rise to a number of 4-regular planar graphs - by regular projections onto the plane - which just have to be enriched by an over/under flag for every vertex to be able to reconstruct the knot from the graph.
What I wonder about:

Question 1: How can I tell which 4-regular planar graphs are possible knot graphs
  (neglecting the flag)? How are knot graphs characterized? Just like
  polyhedral graphs are characterized as exactly the 3-vertex-connected planar
  graphs?

One necessary condition is that the 4-regular planar graph has an Eulerian cycle (which it has in any case) which visits every vertex exactly twice.

Question 2: Is this condition sufficient?

[Added: I suspect that every Eulerian cycle of a 4-regular planar graph has to visit every vertex exactly twice, which means that every 4-regular planar graph fulfills the necessary condition. This implies that Question 2 reads "Is every 4-regular planar graph a knot graph?" Which I did not want to ask, originally.]
 A: I am only going to give a definitive answer to question 2, and perhaps this is what Gerry was getting at.  The answer is no, there are four regular planar graphs, with Eulerian Circuits which are not knots. My example, however, is a link, so it is not a resounding no.  Consider the Hopf link. 

Here is an Eulerian circuit on the corresponding graph.

So, I think we might be able to enforce a condition on always taking the "middle" path on our Eulerian circuits, and that might be sufficient, or at least eliminate examples like this one. But I do not know enough graph theory to say how you would nicely characterize taking the middle path.
Also, do we need some finite condition, otherwise the infinite grid of the integer points in the plane might work?  Or do Eulerian circuits imply finite?
A: If G is a plane, 4-valent, 2-connected graph where all of its faces have a number of sides exactly divisible by 3, then G can't be the projection of a knot. For more details see: Jeong, Dal-Young. "Realizations with a cut-through Eulerian circuit." Discrete Mathematics 137.1 (1995): 265-275.
