Does the elementary knot move really preserve the orientation?

Question

So in this picture, the first diagram changed to the third diagram by the elementary knot moves, but the orientations of the first and the third are different. I wonder if in $R^3$ the moves don't preserve the orientation?

Answer 1

Olivier Bégassat's comment is flawless, and OP agrees that it answers the question, so I am posting it as an answer. (M. Bégassat, if you prefer to post it yourself, please do, and I will delete this and upvote yours.)

You can get from the triangle on the left to that on the right by a rotation $\in SO(3)$, which representes the pinnacle of orientation preservation. So I don't think there is a problem. Maybe you mean the fact that those triangles don't induce the same orientation on the plane they are drawn on, but an orientation on a vector space ($\Bbb R^3$ here) doesn't induce one on its subspaces (for instance the two dimensional plane that carries both triangles), so that would also be a moot point.