Does somebody have a reference (or an argument why it should be true) for the following statement?


“Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 \to S^3$ which takes $K$ to $K'$, then there is also an ambient isotopy $\eta : (S^3,K) \to (S^3,K')$.”


It seems to me that this statement is *always* assumed when citing the Gordon–Luecke theorem from [Gordon, Luecke: Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989), 371-415].


Namely, the authors define knot equivalence by the existence of an $h$ as in the statement (actually they work with homeomorphisms $h$ that are not necessarily orientation-preserving, but they say that everything still works in the orientation-preserving case), while everybody else seems to use the ambient-isotopy definition, even when citing the Gordon–Luecke theorem.



Reference for a fact (?) on homeomorphic knot complements
The result they use is Moise's theorem:


Moise, Edwin E. (1952), Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Annals of Mathematics. Second Series 56: 96–114,


This states that a 3-manifold has a unique PL structure, and a unique smooth structure. 


Moise's theorem was considered so standard back then I imagine Gordon and Luecke didn't bother to cite it because it was so omni-present. 


So the idea goes like this, your homeomorphism can be promoted to a PL-homeomorphism of the sphere. Looking at Moise's paper this is perhaps not clear. The idea is to remove a small tubular neighbourhood of your knot (which you know you can do, since we assume the knot is tame). This gives you a homeomorphism of the knot exterior which preserves the longitude and meridian. This homeomorphism of the exterior can now be promoted to a PL homeo, also a diffeo on the exterior which is isotopic to the original homeomorphism -- so it also preserves the peripheral structure. Now you can extend this map to a PL homeo or a diffeo of the 3-sphere. Since it is orientation-preserving you can apply the Alexander trick to isotope your PL homeo to the identity (through PL-homeomorphisms). 


You could state this as the theorem that the group of orientation-preserving PL homeomorphisms of $S^3$ is connected. This argument was considered standard back then, too. 


If you want to go one further step as Misha suggests, you have to smooth your manifold. Moise's theorem covers that step. But proving that the group of orientation-preserving diffeomorphisms of $S^3$ is connected is harder. That's Cerf's theorem. In Hatcher's answer he mentions that the orientation-preserving group of diffeomorphisms acts trivially on the isotopy classes of knots. Strangely enough, this is an observation that's known to be true for all knots in all homotopy spheres, except perhaps in dimension $4$. The key part of the argument is that homotopy spheres can be decomposed as the union of two standard discs along their boundary. In dimension four, homotopy spheres might have more complicated handle decomposition. 


J.Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ4=0), Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York 1968.


Although Hatcher proved more (and you can in principle use his techniques to prove Cerf's result), Cerf's argument gives a very nice general technique. It is the birthpoint of "Cerf Theory" meaning studying 1-parameter families of smooth functions, showing they can be assumed to be Morse at all but finitely-many times, and describing the cubic singularities where the family fails to be Morse.


