  The expectation value of a Wilson loop in a Chern--Simons theory is a knot
invariant. Its skein relations have been derived in a variety of ways,
including variational methods in which small deformations of the loop are made
and the changes evaluated. The latter method only allowed to obtain approximate
expressions for the skein relations. We present a generalization of this idea
that allows to compute the exact form of the skein relations. Moreover, it
requires to generalize the resulting knot invariants to intersecting knots and
links in a manner consistent with the Mandelstam identities satisfied by the
Wilson loops. This allows for the first time to derive the full expression for
knot invariants that are suitable candidates for quantum states of gravity (and
supergravity) in the loop representation. The new approach leads to several new
insights in intersecting knot theory, in particular the role of non-planar
intersections and intersections with kinks.
