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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.

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    $\begingroup$ If your input data is a knot diagram, then ostensibly you have an "explicit" parametrization of a curve in $\mathbb R^2$ (depending on what you mean by explicit). Place $\mathbb R^2$ on the $xy$-plane in $\mathbb R^3$, then add small perturbations in the $z$-direction near the crossings. Use small normal disks or scale the whole thing up so that unit disks fit in. $\endgroup$
    – Ethan Dlugie
    Commented Mar 8, 2023 at 1:30
  • $\begingroup$ Thistlethwaite and Hoste's program Knotscape will turn a combinatorial description of any knot diagram (Dowker notation) into a planar picture, which can then be turned into a 3d parameterization a la Dlugie's comment. I don't remember if it can handle links. $\endgroup$
    – Jim Conant
    Commented Mar 8, 2023 at 2:34
  • $\begingroup$ One reasonably-efficient way to generate these things would be to enumerate link diagrams then test them for triviality. The test for a link to be trivial is in the Regina software package, for example. It's a combination of the connect-sum decomposition plus Haken's algorithm. Perhaps there are better ways to do what you want, but one nice feature of this construction is it does not come with an explicit way to trivialize the knot. $\endgroup$
    – Ryan Budney
    Commented Mar 8, 2023 at 4:19
  • $\begingroup$ I think you can readily produce an explicit polynomial approximation from a diagram via Bernstein polynomials. i.e. I'm asserting the gap between diagrams and explicit polynomials is small. en.wikipedia.org/wiki/Bernstein_polynomial $\endgroup$
    – Ryan Budney
    Commented Mar 8, 2023 at 5:51
  • $\begingroup$ If $f_1$ is $\sin(\theta)$ and $f_2,f_3$ are sufficiently random (and so that the map is injective), then it will look like a mess when viewed along the $x$-axis. $\endgroup$
    – Ian Agol
    Commented Mar 10, 2023 at 4:49

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