How would you go about proving the $(p, q, r)$ -pretzel knot is equivalent to the $(p, r, q)$ -pretzel knot?
By "equivalent" I mean you can change one knot into the other by elementary deformations.
I've found this question/example in several books and papers on knot theory where they state the proof as obvious.
You can change the $(p,q,r)$ pretzel link into the $(r,q,p)$ pretzel link by a 180 degree rotation around the midpoint of the central twist box (the one containing $q$ twists).
There is also a way to change the $(p,q,r)$ link into the $(q,r,p)$ link. This is a bit three-dimensional, so takes a bit more work to see. Draw the diagram on a sphere $S$ in $R^3$ instead of on a plane in $R^3$. Now you can drag the twist box containing $r$ twists around the back of $S$ ("past infinity") until it becomes the first twist box. (The arcs connecting the twist boxes rearrange themselves in the most convenient way!)
Composing these two operations correctly will move you between the two diagrams. To get a sequence of Reidemeister moves, project all of these movements to a fixed plane.