  The first paper of this series introduced objects (elements of twisted
relative cohomology) that are Poincar\'e dual to Feynman integrals. We show how
to use the pairing between these spaces -- an algebraic invariant called the
intersection number -- to express a scattering amplitude over a minimal basis
of integrals, bypassing the generation of integration-by-parts identities. The
initial information is the integrand on cuts of various topologies, computable
as products of on-shell trees, providing a systematic approach to generalized
unitarity. We give two algorithms for computing the multi-variate intersection
number. As a first example, we compute 4- and 5-point gluon amplitudes in
generic spacetime dimension. We also examine the 4-dimensional limit of our
formalism and provide prescriptions for extracting rational terms.
