TITLE: Tying knots via gravity-assisted spaceship trajectories
QUESTION [22 upvotes]: Q.
  Can every knot be realized as the trajectory of a spaceship
  weaving among a finite number of fixed planets, subject to gravity alone?


          


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.

          


          

Cassini gravity-assist trajectory. 
Image from NASA/JPL.

REPLY [8 votes]: Yes, with some caveats. For an authoritative source, please see the monograph Koon et al. "Dynamical systems, the three-body problem and space mission design"
The short story is that with 2 or more large bodies, trajectory of a spacecraft is "chaotic", and hence under some conditions, it can be shown that horseshoe-type dynamics exist. In other words, if you label the regions around each large body with an alphabet, any arbitrary string of alphabets can be achieved"
Also see: http://www2.esm.vt.edu/~sdross/papers/AmericanScientist2006.pdf