  We compute the non-MHV one-loop seven-gluon amplitudes in N=4
super-Yang-Mills theory, which contain three negative-helicity gluons and four
positive-helicity gluons. There are four independent color-ordered amplitudes,
(- - - + + + +), (- - + - + + +), (- - + + -+ +) and (- + - + - + +). The MHV
amplitudes containing two negative-helicity and five positive-helicity gluons
were computed previously, so all independent one-loop seven-gluon helicity
amplitudes are now known for this theory. We present partial information about
an infinite sequence of next-to-MHV one-loop helicity amplitudes, with three
negative-helicity and n-3 positive-helicity gluons, and the color ordering (- -
- + + ... + +); we give a new coefficient of one class of integral functions
entering this amplitude. We discuss the twistor-space properties of the
box-integral-function coefficients in the amplitudes, which are quite simple
and suggestive.
