I have a somewhat empirical question which I hope is still welcome here. I would like to know how to write down explicit parameterizations of "complicated unlinks", say with 2 or 10 components, in $\mathbb{R}^3$ (where complicated here means roughly, if you look at it, it's a big mess).

Perhaps a strategy to obtain such parameterizations would involve starting with a standard simple parameterization and post-composing with a series of complicated, yet explicit, homeomorphisms of $\mathbb{R}^3$.

For what it's worth, I have one more hope for these parameterizations, namely that the tubular neighborhood given by looking at the unit disk bundle of the image disjoint union of $S^1$'s (with the usual metric on $\mathbb{R}^3$) should be embedded. In other words, the different components should not get too close to one another or themselves.

Any thoughts on how to obtain such parameterized unknots or examples of such unknots?

(Here's a fun diagram also in this question. This is the sort of thing I'd explicit parameterizations of.)

edit: Thanks for the input everyone. By explicit, I really mean explicit - I want to use this for some computer experiments. I very much was hoping not to go by way of diagrams since I don't know how to turn those into explicit equations very easily. I'm really looking for a couple of maps \begin{align} S^1 &\to \mathbb{R}^3 \\ \theta &\mapsto (f_1(\theta), f_2(\theta), f_3(\theta)) \end{align} one for each component, where the functions $f_i$ are something explicit enough that I can reasonably approximate them on my computer.