  We study a complex analogue of a Wilson Loop, defined over a complex curve,
in non-Abelian holomorphic Chern-Simons theory. We obtain a version of the
Makeenko-Migdal loop equation describing how the expectation value of these
Wilson Loops varies as one moves around in a holomorphic family of curves. We
use this to prove (at the level of the integrand) the duality between the
twistor Wilson Loop and the all-loop planar S-matrix of N=4 super Yang-Mills by
showing that, for a particular family of curves corresponding to piecewise null
polygons in space-time, the loop equation reduce to the all-loop extension of
the BCFW recursion relations. The scattering amplitude may be interpreted in
terms of holomorphic linking of the curve in twistor space, while the BCFW
relations themselves are revealed as a holomorphic analogue of skein relations.
