If we take a knot diagram and ignore the over/under-crossing information, we obtain a shadow (or projection). Thus a shadow is simply a plane embedding of a 4-regular graph. Clearly, two non-equivalent knots can give rise to the same shadow: in the figure below we have a knot diagram of the trefoil knot (left) and of the unknot (center), which yield the same shadow (right).
Going in the opposite direction, a shadow with $n$ vertices induces $2^n$ knot diagrams, since we can assign to each vertex an overcrossing or an undercrossing.
In this same forum, Joe O'Rourke asked if at least one of these $2^n$ knot diagrams corresponds to the unknot (the answer is yes).
My question is:
Start with a shadow of $n$ vertices. How many different knot types can be obtained from the $2^n$ knot diagrams induced by the shadow?
For instance, for the shadow on the right hand side of the figure above, the answer is $3$: by assigning overcrossings or undercrossings to each vertex, one can only obtain the trefoil knot, its mirror image, and the unknot.
The answer surely depends on the shadow. Andy Putman's answer highlights an important point (which was also observed by Lou Kauffman in an email message) : one should only consider shadows with no "cut vertices", since such vertices will become nugatory crossings in a knot diagram. If we don't impose this restriction, there are examples such as the one below (Andy's example). No matter what assignment of over/under-crossings this shadow receives, each crossing will be nugatory, and the corresponding knot will always be the unknot:
So one should restrict the question to shadows with no cut vertices.
Lou Kauffman has observed that in this case, the two alternating diagrams associated to the shadow correspond to non-trivial knots. This follows since that the number of crossings in an alternating, reduced diagram is an invariant of the alternating knot (for a proof, see http://homepages.math.uic.edu/~kauffman/Bracket.pdf). Thus a partial answer for a shadow with no cut vertices is: at least two of the diagrams it induces correspond to a non-trivial knot.
My guess is that the answer is "exponentially many'' for "most" shadows. Or at least for some? More specifically: is it true that there exists a $c>1$ such that for all sufficiently large $n$, there exists a shadow whose $2^n$ associated diagrams yield at least $c^n$ different knot types?
I have spent quite a bit of time searching for pretty much any result in this direction, without any success.