According to this article, every knot can be realized as the intersection of $2$ non-singular algebraic sets in $\Bbb{R}^4$, one of which is a standard $\Bbb{S}^3$. However this result use many ambient conditions, so I want to know if these conditions can be restricted, namely:
Does there exist an algebraic morphism from $\Bbb{R}\Bbb{P}^1$ to $\Bbb{R}\Bbb{P}^3$ that is a topological embedding but not isotopic to the standard line in $\Bbb{R}\Bbb{P}^3$?
and a question of opposite direction:
Is every smooth embedding from $\Bbb{R}\Bbb{P}^1$ to $\Bbb{R}\Bbb{P}^3$ isotopic to the image of an algebraic morphism from $\Bbb{R}\Bbb{P}^1$ to $\Bbb{R}\Bbb{P}^3$?
A complex version can also be stated by replacing $\Bbb{R}$ with $\Bbb{C}$. Due to the rigidity of complex structures, this version probably has a negative answer but I can not prove it.