is knot type invariant under diffeomorphism?

Question

Is it possible to have a diffeomorphism of $R^3$ which changes the knot type, for instance the image of a trivial knot is a trefoil knot?

Answer 1

It is well known that the reflection of a trefoil knot is not the same type as the trefoil knot itself. On the other hand any diffeomorphism which preserves orientation can be interpolated to the identity through an isotopy... hence it preserves knot type. Instead diffeomorphism which invert orientation would change the type of chiral knots (as the trefoil knot) but not achiral knots (like the trivial knot).

Answer 2

Just to state that the answer given here by Emanuele depends on the fact that the isotopy classes of self-homeomorphisms of $\mathbb R^n$ , which is {$Id,-Id$}. If your not is embedded in a different space X, the answer will depend on the isotopy classes of homeomorphisms of X.

Basically, we want to see if , for h a homeomrphism , whether, $K=Id(K)$ and $h(K)$ are isotopic. But since MCG$(\mathbb R^n)= {\pm Id} $ , we have that h(K) is isotopic to $\pm Id K= \pm K$ .