Reference for a fact (?) on homeomorphic knot complements

Question

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.

Answer 1

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.

Answer 2

This is a question that I remember worrying about when I first started learning about knot theory. Older books have a tendency to skim over this point rather lightly, perhaps because the resolution of the question seems to involve techniques that have little to do with standard knot theory. One book that doesn't avoid treating this issue squarely is Burde and Zieschang's "Knots" (now in a third edition with a third author, Heusener, and available in an electronic version). In Chapter 1 of this book the fact that every orientation-preserving homeomorphism of $S^3$ (or equivalently ${\mathbb R}^3$) is isotopic to the identity is attributed to a 1960 paper of G.M.Fisher. (In Kirby's landmark 1969 paper on the Stable Homeomorphism Conjecture he refers to this result just as being "classical".)

The earlier work of Moise on existence and uniqueness of PL structures on 3-manifolds does not seem to say anything about homeomorphisms between PL 3-manifolds being isotopic to PL homeomorphisms. Moise just proves that such homeomorphisms can be approximated by PL homeomorphisms. However, one of his approximation theorems appears to be sufficient to deduce the desired fact about knots. Namely, suppose $f:S^3\to S^3$ is an orientation-preserving homeomorphism taking a knot $K$ to a knot $K'$. Choose an open set $U$ in $S^3-K$ and choose a continuous function $\varepsilon: U \to (0,1)$ that approaches $0$ at the frontier of $U$. Then Moise's theorem states that $f$ can be $\varepsilon$-approximated by a homeomorphism $g$ which is PL on any compact set in $U$ (which just means PL on $U$ itself) and which equals $f$ outside $U$. In particular $g$ takes $K$ to $K'$. Since $g$ is PL on some open set, it can easily be isotoped (topologically) to the identity, and this isotopy restricts to a topological isotopy of $K'$ to $K$. If $K$ and $K'$ are PL knots, one might want a PL isotopy, but this argument doesn't give it.

The Burde-Zieschang book does prove that two PL knots that are equivalent via a topological homeomorphism that preserves orientation are PL isotopic. This is Corollary 3.17, and they deduce it from Waldhausen's theorem that, in the PL category, two knots are isotopic if they have isomorphic "peripheral systems". The latter are defined in terms of the fundamental group of the knot complement, so they are invariant under topological homeomorphisms preserving orientation.

I don't know who first proved the general result that every homeomorphism between PL 3-manifolds is isotopic to a PL homeomorphism, but there is a very nice proof using just PL topology of 3-manifolds and the Kirby torus trick in a 1976 paper of A.J.S.Hamilton.

In the smooth category Cerf's theorem ($\Gamma_4=0$) certainly implies that if two smooth knots are equivalent under an orientation-preserving diffeomorphism then they are smoothly isotopic. However, one can obtain this without using Cerf's theorem by an elementary argument that works in all dimensions, even when the analog of Cerf's theorem fails (due to the existence of exotic spheres). Suppose $f:S^n\to S^n$ is an orientation-preserving diffeomorphism taking a closed proper subset $K\subset S^n$ to another such set $K'$. Since $f$ preserves orientation, it is a standard fact that $f$ can be isotoped to be the identity on a ball $B\subset S^n-K$, so we assume this has been done. Another way of stating this standard fact is to say that the map from the diffeomorphism group $Diff(B,\partial B)$ of $B$ fixing (a neighborhood of) $\partial B$ to the orientation-preserving diffeomorphism group $Diff^+(S^n)$ of $S^n$, obtained by extending diffeomorphisms via the identity outside $B$, induces a surjection on $\pi_0$. This means that we can compose $f$ with a diffeomorphism $S^n\to S^n$ supported in $B$ representing $[f]^{-1} \in \pi_0Diff^+(S^n)$ to get a new diffeomorphism that is isotopic to the identity and still takes $K$ to $K'$. (This can be done quite explicitly in fact.)

[The second half of this last paragraph has been revised to correct a misstatement.]