Knot invariants as nondegenerate quantum geometries

The loop-space representation based on Ashtekar’s new variables has allowed for the first time the construction of quantum states of the gravitational field. However, all states known up to the present were associated with spacetime metrics that were everywhere degenerate. In this Letter we present a new exact solution of the constraint equations of quantum gravity that is the first quantum state of the gravitational field known to be associated with a not-everywhere-degenerate metric. The state is associated with the second coefficient of the Alexander-Conway polynomial of knot theory.