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So I saw that you get a Hilbert space spanned (in some sense) by some kind of labelled knotted graphs (up to diffeomorphism) in a manifold. And I see how they could be described combinatorially in some way.

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Yes, these go by various names depending on how many of the constraints they solve. Cylindrical functions, spin networks (solve the Gauss constraint), s-knots also sometimes called "abstract spin networks" (solve the diffeomorphism constraint), etc.

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But then if you stick in a surface, you Hilbert space seems to get bigger. For each knotted graph you had before, know you have many corresponding to all the different (up to whatever appropriate equivalence) ways the knotted graph can intersect the surface. Then you can stick more and more surface in, and get bigger and bigger Hilbert spaces, until you need every knotted graph (no longer up to diffeomorphism) on your original manifold.

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I am not sure what you're talking about here, but I am going to guess that you're talking about something like calculations of black hole entropy, which does involve a background spacetime and inserting a surface corresponding to the black hole horizon, and getting a Hilbert space of "surface geometry" from the spin networks. This is certainly an approximation -- the full quantum state of the spacetime isn't known, but putting in reasonable boundary conditions allows you to calculate the entropy anyway. In this case, the manifold on which you play (and surfaces you consider on it) is indeed supposed to be taken as physical spacetime -- a semiclassical approximation to the full quantum state. This is simply because it is too hard to write out the full quantum state -- this is somewhat analogous to QFT calculations of the EM field inside a conducting cavity. The full EM field, taking into account the atomic structure of the metal isn't explicitly calculated because that is too hard to write down -- instead, reasonable boundary conditions are written down, and the much simpler problem of just the fields in the cavity can be considered.

But this doesn't mean that the formalism in general depends fundamentally on a background spacetime manifold -- it does not. However, introducing one as an approximation can make certain calculations of interest many orders of magnitude easier/more practical to compute.

I am not exactly sure this is what you referring to, but hopefully it illustrates the point that you are worrying about...