  Let $K$ be a knot in the 3-sphere, viewed as the ideal boundary of hyperbolic
4-space $\mathbb{H}^4$. We prove that the number of minimal discs in
$\mathbb{H}^4$ with ideal boundary $K$ is a knot invariant. I.e.\ the number is
finite and doesn't change under isotopies of $K$. In fact this gives a family
of knot invariants, indexed by an integer describing the extrinsic topology of
how the disc sits in $\mathbb{H}^4$. These invariants can be seen as
Gromov--Witten invariants counting $J$-holomorphic discs in the twistor space
$Z$ of $\mathbb{H}^4$. Whilst Gromov--Witten theory suggests the general scheme
for defining the invariants, there are substantial differences in how this must
be carried out in our situation. These are due to the fact that the geometry of
both $\mathbb{H}^4$ and $Z$ becomes singular at infinity, and so the
$J$-holomorphic curve equation is degenerate, rather than elliptic, at the
boundary. This means that both the Fredholm and compactness arguments involve
completely new features.
