. Can every knot be realized as the trajectory of a spaceship weaving among a finite number of fixed planets, subject to gravity alone?Q

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.)

. 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?Q'

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.

^{ Cassini gravity-assist trajectory. Image from NASA/JPL. }