Take a thick cord (the alim cord of your laptop or your mouse cord for example) and wrap it around your hand (or finger) turning always in the same direction but possibly knotting it. Then try to deform it using only the "obviously allowed moves", for exemple the first Reidemeister move is not allowed because it is difficult to perform it without "cheating".
More precisely, let's call a round knot a knot diagram which does not pass through $0$ and where the angle (in polar coordinates) of a point moving along the knot is always increasing.
I'm not sure of what should be the precise condition for two knots to be equivalent. For example I want to allow the second Reidemeister move, even if the result is not round anymore (but at the end, we must finish on a round knot of course). The idea is that only more or less planar moves are allowed, and Reidemeister 1 is not allowed because it involves untwisting a loop which is not really planar. Perhaps there are also new moves to be added, I'm not sure.
For example the torus knots are obviously round knots (and the torus knots with one parameter being 1 are trivial as usual knots but not at round knots), and the figure eight knot is also a round knot (and I haven't found a knot that cannot be represented as a round knot yet). There should also be an invariant which is the number of times the knot turns around 0.
Is there anything that can be said about these knots?
Do you have a more precise definition? Have they been studied? Are they classified?