Let K be a knot or link in the 3-sphere, thought of as the ideal boundary of hyperbolic 4-space H^4. I will describe a programme to count minimal surfaces in H^4 which have K as their asymptotic boundary. This should give an isotopy invariant of the knot. I will explain what has been proved and what remains to be done. Minimal surfaces actually correspond to J-holomorphic curves in the twistor space Z->H^4 (a certain S^2 bundle over H^4) and so this invariant can be seen as a Gromov-Witten type invariant of Z. The big difference with “standard” situations is that the almost complex structure on Z (equivalently, the metric on H^4) blows up at the boundary. This means the J-holomorphic equation, or minimal surface equation, becomes degenerate at the boundary of the domain. As a consequence, both the Fredholm and compactness parts of the story need to be reworked by hand. If there is time I will explain how this can be done, relying on results of Mazzeo-Melrose from the 0-calculus, and also some results from the theory of minimal surfaces.