TITLE: How to calculate the Witten-Reshetikhin-Turaev invariants from a triangulation?
QUESTION [17 upvotes]: I'm interested in the Witten-Reshetikhin-Turaev invariants for 3-manifolds, and in particular, how to calculate them from a triangulation of the 3-manifold (recall that as they were first introduced, their calculation is based on a surgery diagram for the manifold).  Is there any some sort of "state sum" model giving the WRT invariant for a $3$-manifold, which is based on the triangulation of the $3$-manifold?
I am well aware of the Turaev-Viro construction, and it is exactly the type of definition I'm hoping for, but of course the TV invariant is $\left|\cdot\right|^2$ of the WRT invariant, not the WRT invariant itself.
The simpler the construction, the better.  Certainly one can go algorithmically from a triangulation to a surgery diagram, but I want something which is conceptually based on the triangulation.
I guess any answer I get will also include the case for a pair $(Y^3,L)$, and even perhaps allow for $M^3$ to have nonempty boundary (i.e. the full tqft).  However, I will be satisfied  even if I just get a construction for a closed $3$-manifold $M$.

REPLY [3 votes]: Hi, we (Ricardo Machado and Sostenes Lins) put in arxiv, 3 articles which shows how to get a framed link presentation from a triangulation. We show an O(n^2) algorithm which I implemented and it is really working. So with the framed link we can calculate the WRT-invariants 
The link to the articles:
http://arxiv.org/find/all/1/all:+AND+lins+AND+sostenes+AND+ricardo+Machado/0/1/0/all/0/1