In algebraic topology course, we learnt the Alexander's horned sphere, an example of not all embeddings of a 2-sphere into the 3-sphere is not isotopic to the standard one.
Thus one can say that the Alexander horned sphere is an "exotic" embedding.
I'm curious that the same is true for an embedding of a disk into a sphere; i.e., there exists an embedding of a $k$-disk into $n$-sphere that is not isotopic to the standard embedding. (here I mean the usual embedding that has contractible complement, for instance the upper half sphere inside a sphere)
I doubt the existance of such embeddings(just as my intuition, or my lack of imagination, says!!), but don't know how to prove it rigorously.
Maybe it's slightly easier to consider the embedding into $R^n$ via the stereographic projection, but still have difficulties.
Also, it's not quite clear to me that the result depends on whether the disk is open or closed. Maybe open disks can be subtle as the complement is not a manifold in general.
Many thanks in advance.
In the smooth world the answer is no. This is usually called "the Disk theorem" https://en.wikipedia.org/wiki/Disc_theorem and is due to Palais.
In the topological world things are more fuzzy. If the disk is not assumed to be locally flat (in other words each point has a neighborhood where after a homeomorphism the copy of $D^k$ is a linear subspace of $\Bbb R^n$) then there are examples. For instance there are embeddings of the cone on a knot in $S^4$, which gives a 2-disk inside of $S^4$ which is not isotopic (smoothly or even topologically) to the standard embedding.
I believe if the embedding is locally flat there exists a neighborhood of it which has a smooth structure where it is diffeomorphic to the standard disk bundle over the disk and you can use Palais' theorem to construct an isotopy between that and the standard embedding, but I haven't worked this out carefully.
There are also dimensional considerations which imply all topological embeddings are automatically locally flat .