  We investigate the Ponzano-Regge and Turaev-Viro topological field theories
using spin networks and their $q$-deformed analogues. I propose a new
description of the state space for the Turaev-Viro theory in terms of skein
space, to which $q$-spin networks belong, and give a similar description of the
Ponzano-Regge state space using spin networks.
  I give a definition of the inner product on the skein space and show that
this corresponds to the topological inner product, defined as the manifold
invariant for the union of two 3-manifolds.
  Finally, we look at the relation with the loop representation of quantum
general relativity, due to Rovelli and Smolin, and suggest that the above inner
product may define an inner product on the loop state space.
