  The discrete picture of geometry arising from the loop representation of
quantum gravity can be extended by a quantum deformation. The operators for
area and volume defined in the q-deformation of the theory are partly
diagonalized. The eigenstates are expressed in terms of q-deformed spin
networks. The q-deformation breaks some of the degeneracy of the volume
operator so that trivalent spin-networks have non-zero volume. These
computations are facilitated by use of a technique based on the recoupling
theory of SU(2)_q, which simplifies the construction of these and other
operators through diffeomorphism invariant regularization procedures.
