Is the problem obvious for the closure of open unbounded subsets of $\mathbb{R}^3$?

@PietroMajer Yes we may assume $U=V=(0,1)$

but the torus admit both continuous and discrete(Arnold cat map) hyperbolic dynamics

@PierrePC To be honnest after the comment of Yves I thought that perhaps your answer do. But I realized that you perturbe the initial system. So a knot type can be realized by an algebraic system not the original knot. Agree?

@YCor and the same point you mentiined but for linki

This is an excuse to search for 2 diffeomorphic submanifolds of $R^n$ with non diffeomorphic tubular neighborhood? Any way I appreciate your beautiful idea

Now i completely understand what you say: a tubular neighborhood of a knot is modeled in an abstract solid torus whose center is a hyperbolic periodic attractor so it survives with algebraic approximation. In fact in your argument you reconstruct the structural stability of hyperbolic attractors. (They survive with small perturbation). BTW is it obvious that a tubular neighborhood of every kknot is diffeomorphic to the solid tori?My guess is yes because they have trivial normal bundle?

yes I see so let me think more about that

@YCor And I guess the answer was negative by some counting(countable-non countable) argument. I think Loic Teyssier answer to my question. any way your question is very interesting. Thanks about that

@YCor Yes that is an interesting question. This remind me of my past (MO or non MO?) as follows: is every closed analytic curve a limit cycle of an algebraic vector field. If I am not mistaken I asked this question in MO