One classical example of such move is [Conway mutation][1], 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][2] 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).

[1]: https://en.wikipedia.org/wiki/Mutation_(knot_theory)
[2]: https://arxiv.org/abs/1103.0487