  An analysis of the action of the Hamiltonian constraint of quantum gravity on
the Kauffman bracket and Jones knot polynomials is proposed. It is explicitely
shown that the Kauffman bracket is a formal solution of the Hamiltonian
constraint with cosmological constant ($\Lambda$) to third order in $\Lambda$.
The calculation is performed in the extended loop representation of quantum
gravity. The analysis makes use of the analytical expressions of the knot
invariants in terms of the two and three point propagators of the Chern-Simons
theory. Some particularities of the extended loop calculus are considered and
the implications of the results to the case of the conventional loop
representation are discussed.
