[j2kun]

The reason I don't think it's so obvious is because, while the input knots may be finite structures, the transformation between the two might be unbounded in size. E.g., one can repeatedly "blow up" a knot presentation by adding in superfluous expansions of identities like L=LL; it might be the case that certain combinations of these trivial identities combine to make some nontrivial manipulations of a knot.

This is essentially the reason why there is no algorithm, which when given a finite group presentation and two input words, can decide whether the two words represent the same group element.