These are the unlink. The problem is that you can push all of the knotting off to infinity.
If you take any $z$ cross-section, you see a collection of points in the plane, say countably many. So you have a map $\varphi: \mathbb{Z}\times \mathbb{R} \to \mathbb{R}^3$ such that $\varphi(n, z) = (f_1(n,z),f_2(n,z), z)$. Now, create an isotopy $\varphi_t$ such that $\varphi_t(n,z)= (f_1(n,0),f_2(n,0), z)$ for $-t\leq z\leq t$, and essentially $\varphi_t(n,z)=(f_1(n,z-t),f_2(n,z-t),z)$ for $z\geq t$, and $(f_1(n,z+t,f_2(n,z+t),z)$ for $z\leq -t$ (this will be non-smooth at levels $t,-t$, but you can easily modify it a bit to be smooth by reparameterizing the $z$ coordinate by a mollifier). As $t\to \infty$, the link becomes isotopic to the straight vertical link $\{(f_1(n,0),f_2(n,0)) | n\in \mathbb{Z} \}\times \mathbb{R}$. You're essentially combing the link to be straight. This can easily be described by ambient isotopy of diffeomorphisms.