If I understand the theory of quantum invariants of 3-manifolds correctly (possibly I don't), different presentations of 3-manifolds produce different TQFTs but the same quantum invariants (Reshetikhin-Turaev invariants for example). These invariants are said to be analogous to observables in quantum field theory. Do the TQFTs have a physical analogy?
EDIT: While trying to be more specific in my example below (per a commenters request), I realized I was thinking about TQFTs incorrectly and that my original question didn't make much sense. However, a new question fell out when I tried to clarify below.
Here is my attempt to be more specific via an example and my new question: Take for example this video at 1:00:50. The speaker gives a presentation in terms of generators of a 3-dimensional TQFT for $\mathbb{R}P^3$ using Heedgaard decomposition as a tool. He then calculates the value produced by the field theory on the generators in his presentation of $\mathbb{R}P^3$. There are other Heedgard decompositions and presentations of $\mathbb{R}P^3$ which will give a different value when you apply the field theory to the generators. However, the Reshetikhin-Turaev invariant should be the same for all presentations of $\mathbb{R}P^3$. I read somewhere (can't remember exactly where) that the Reshetikhin-Turaev invariant corresponds to an observable in Physics. Does the value of the field theory prior to calculatng the RT invariant have a physical interpretation?