Every 1-knot has a sort of 2-knot analog (actually, a whole family of them) made by "revolving" the knot, but there are 2-knots not generated in this manner. (The same holds true for higher dimensional knots.)
As we go higher, there are knot moves not even possible in lower dimensions (such as twists).
Just a thought.
The Jones polynomial is a thing which can tell knots apart. The quantum computer can calculate the Jones polynomial by having particles trace out the knot. The Jones polynomial usually takes exponential time to calculate.
The "topological" in "topological quantum computing" refers to the braiding and the fact that they're computing topological invariants of a space.
Edit. For the very bravest of five year olds, I wrote a paper on these things here: https://arxiv.org/abs/1610.05384