If one imagines turning a knot, and looking at its projection on the plane, it will change between different projections of the same knot. In between, there will be some singularities when more then 2 points get projected to the same point on the plane.
Not all such transformations of the projection correspond to Reidemeister moves, for example turning the unknot, represented by a circle in 3-space, gives a singularity where all points all projected onto a line, before it is the unknot again.
However, this could be prevented by modifying the knot a bit by an isotopy, so that this turning would correspond to some Reidemeister moves.
My question is: Can every knot be slighly modified in some way so that only Reidemeister moves happen? Or formulated in another way: Are there some exotic moves that can happen to the projection of a knot if I rotate it?
I hope I expressed myself clearly, even if I am not sure if this question makes sense.
So I have no proof, but I think it is true. The program KnotPlot (free to download and play with) is useful here. You can draw a knot and play with your idea here and check it a lot from different angles and things.
Using general position, you can assume that you have wiggled your knot a little everywhere, so that you only ever have triple points show up in your rotation ( which correspond to a type III move) in a finite number of angles, having chosen an axis of rotation. Similarly, there will be a finite number of double points appear from type II moves. Type I moves are a little weirder, I think where your view is rotating through a line tangent to knot.
I think the proof to what you are asking might end up being somehow equivalent to Reidemeister's Theorem, but again, I am not positive here.
Here are two images from KnotPlot I made that are almost a single Reidemeister type III move. I seem to have picked up a type I move too, but if you are careful, you can rotate just enough to make a single move.
You should make your knot as thin as reasonable here. Thicker models will make it unclear what moves are happening in what order.