I'm currently learning about knot theory for the first time from Basic Topology by Armstrong, and he states that two knots $k_1, k_2\subset \mathbb{R}^3$ are equivalent if there exists a homeomorphism $h: \mathbb{R}^3\to\mathbb{R}^3$ such that $h(k_1) = k_2$.
My question is: why is it a homeomorphism from $\mathbb{R}^3$ to $\mathbb{R}^3$, instead of $h: k_1\to k_2$?
Are these statements equivalent, because if I restrict $h: \mathbb{R}^3\to\mathbb{R}^3$ to just $k_1$, and $h^{-1}$ to just $k_2$, then wouldn't this be a homeomorphism between $k_1$ and $k_2$?
I could imagine that it has something to do with that another way to define them, knots are not defined as isotopies, but as ambient isotopies. Otherwise, you could "pull at the ends", such that the holes become infinitely small and disappear.
In other words: It could help to preserve the knot's holes within $\mathbb{R}^3$. (Note that it has no holes as a subspace, therefore the disappearance of holes when seen as embedding in $\mathbb{R}^3$ does not contradict that homeomorphisms retain the topology between the subspaces.)