One classical example of such move is Conway mutation, which falls into the category of tangle replacement, as Qiaochu Yuan mentioned in his comment. There's a very famous pair of mutants, the Kinoshita-Terasaka and the Conway knot (see the Wikipedia article).
Apparently, there's some topology behind this move: recently, using knot Floer homology, Josh Greene has shown that two alternating knots are mutants if their branched double covers are homeomorphic, and the other arrow was shown by Viro (see references in Greene's paper).
