  We develop a systematic approach for proving the existence of choreographic
solutions in the gravitational $n$ body problem. Our main focus is on spatial
torus knots: that is, periodic motions where the positions of all $n$ bodies
follow a single closed which winds around a $2$-torus in $\mathbb{R}^3$. After
changing to rotating coordinates and exploiting symmetries, the equation of a
choreographic configuration is reduced to a delay differential equation (DDE)
describing the position and velocity of a single body. We study periodic
solutions of this DDE in a Banach space of rapidly decaying Fourier
coefficients. Imposing appropriate constraint equations lets us isolate
choreographies having prescribed symmetries and topological properties. Our
argument is constructive and makes extensive use of the digital computer. We
provide all the necessary analytic estimates as well as a working
implementation for any number of bodies. We illustrate the utility of the
approach by proving the existence of some spatial choreographies for $n=4,5,7$,
and $9$ bodies.
