Is the third coefficient of the Jones knot polynomial a quantum state of gravity?

Some time ago it was conjectured that the coefficients of an expansion of the Jones polynomial in terms of the cosmological constant could provide an infinite string of knot invariants that are solutions of the vacuum Hamiltonian constraint of quantum gravity in the loop representation. Here we discuss the status of this conjecture at third order in the cosmological constant. The calculation is performed in the extended loop representation, a generalization of the loop representation. It is shown that the Hamiltonian does not annihilate the third coefficient of the Jones polynomial (${\mathit{J}}_3$) for general extended loops. For ordinary loops the result acquires an interesting geometric meaning and new possibilities appear for ${\mathit{J}}_3$ to represent a quantum state of gravity. © 1996 The American Physical Society.