Here are my two questions:
Given a finite connected non-oriented planar graph, is there a way to determine whether or not it is possible to derive a single non-trivial knot diagram from this graph, so that every graph edge corresponds to a single knot crossing? (The solution should be other than traversing through every possible knot combination for this graph.) In other words, how to determine the minimal number of individual link components that an arbitrary graph can carry? (Should I stick to a particular knot polynomial to solve this task?)
Is there any dependency between certain graph families and the existence of such knot? (For example, do multigraphs need special treatment? Or, may the Eulerian/Hamiltonian graphs be suggestive? What about bipartite and line graphs?)
Thank you.