Imagine that I have a closed or open piece of string that I move continuously through space without self-intersection. Formally, this means that I have a homotopy $\phi_t:[0,1]\times X\to\Bbb R^3$ where $X=[0,1]$ or $X=S^1$ and so that $\phi_t$ is a homeomorphism onto its image for all $t\in[0,1]$.
Question: Does this homotopy extend to a homotopy of the whole space? That is, is there a homotopy $\bar\phi_t:[0,1]\times \Bbb R^3\to\Bbb R^3$ so that $\bar\phi_t$ is a homeomorphism and $\bar\phi_t|_X=\phi_t$ for all $t\in[0,1]$?
Is this a standard result with a name? It strikes me as something belonging to knot theory, but I am not familar enough with this subject to recognize the result.
My main question is stated as above. But I do also wonder how general such an extension property can hold (if at all). For example, I believe it fails for $X=S^2$ (Alexander's horned sphere maybe?).