> ***Q***.
Can every knot be realized as the trajectory of a spaceship
weaving among a finite number of fixed planets, subject to gravity alone?
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<img src="https://i.sstatic.net/O0fXU.jpg" width="200" />
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To make this more specific, let $S$ be a large sphere,
containing a finite number of planet-points $P=\{p_1, \ldots, p_n\}$.
The planet-points have (in general) different
masses, and are fixed in $\mathbb{R}^3$.
A spaceship $x$ approaches $S$ from $\infty$, interacts via gravity
with the point masses, and eventually exits $S$ to $\infty$.
Define the knot $K$ realized by the ship's trajectory as the
path of $x$ plus a connection between the two ends at $\infty$.
(Assume those two $\infty$-ends are distinct.)
> ***Q'***. For any given knot $K$, can one arrange point masses in $P \subset S$ and a line
and speed of approach
to $S$ so that $x$'s path weaves $K$ by interacting with the planets via gravity alone,
i.e., without the use of rocket fuel?
One approach might be to design a "gadget" that mimics a vertex
$v$ of a stick knot and $v$'s two incident segments.
But preventing the vertex gadgets from interfering
with one another might not be straightforward.
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<img src="https://i.sstatic.net/6tKJO.png" width="400" />
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<sup>
Cassini gravity-assist trajectory.
Image from [NASA/JPL](https://saturn.jpl.nasa.gov/resources/1776/cassini-trajectory/).
</sup>
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[1]: https://i.sstatic.net/O0fXU.jpg