  In the context of canonical quantum gravity in terms of Ashtekar's new
variables, it is known that there exists a state that is annihilated by all the
quantum constraints and that is given by the exponential of the Chern--Simons
form constructed with the Asthekar connection. We make a first exploration of
the transform of this state into the spin-network representation of quantum
gravity. The discussion is limited to trivalent nets with planar intersections.
We adapt an invariant of tangles to construct the transform and study the
action of the Hamiltonian constraint on it. We show that the first two
coefficients of the expansion of the invariant in terms of the inverse
cosmological constant are annihilated by the Hamiltonian constraint. We also
discuss issues of framing that arise in the construction.
