  Witten's twistor string theory gives rise to an enigmatic formula
[arXiv:hep-th/0403190] known as the "connected prescription" for tree-level
Yang-Mills scattering amplitudes. We derive a link representation for the
connected prescription by Fourier transforming it to mixed coordinates in terms
of both twistor and dual twistor variables. We show that it can be related to
other representations of amplitudes by applying the global residue theorem to
deform the contour of integration. For six and seven particles we demonstrate
explicitly that certain contour deformations rewrite the connected prescription
as the BCFW representation, thereby establishing a concrete link between
Witten's twistor string theory and the dual formulation for the S-matrix of N=4
SYM recently proposed by Arkani-Hamed et. al. Other choices of integration
contour also give rise to "intermediate prescriptions". We expect a similar
though more intricate structure for more general amplitudes.
