# What's the best reference for actual formulas for RT invariants?

If one really wants to understand the formulas for how to construct the Reshetikhin-Turaev 3-manifold invariants coming from quantum groups in terms of R-matrices and such, what's the best reference to read?

Edit: Apparently I'm being too vague. Let me explain my motivation a little bit. Right now, I'm thinking about how to categorify Chern-Simons theory. I understand most of the maps from quantum groups like the R-matrix quite well, so I would like a good reference that has formulae I can try to categorify based the bits I already understand.

-
Are you interested in the details of the R-matrix and other U_q(g) stuff, or are you more interested in combinatorial topology and handle-slide invariance? Or something else? –  Kevin Walker Oct 13 '09 at 19:54
Or by "actual formulas" do you mean detailed formulas for specific 3-manifolds (e.g. Lens Spaces)? –  Kevin Walker Oct 13 '09 at 19:56
I'm interested in general formulas that I actually understand each term of in terms of quantum groups. Though understandable stuff about combinatorial topology would be good too. –  Ben Webster Oct 13 '09 at 20:51

## 3 Answers

My guess is that you are mainly interested in going from a quantum group to a modular tensor category. If so, then this paper by Steve Sawin is fairly explicit and general.

Probably other people (Noah?) have a much better knowledge of the literature than I do.

-

Since no one else has jumped in, I'll try again.

Standard references producing a TQFT from a MTC include Turaev's big book and the shorter notes of Bakalov and Kirillov (available online here).

If you want something less algebraic and more combinatorial (which I doubt), I like the book by Kauffman and Lins, and also some paper(s) of Lickorish's from the early 1990's.

I haven't read the original Reshetikhin and Turaev papers in a long time, but I remember thinking at the time that they were perfectly readable.

-
The original Reshetikhin and Turaev papers are quite readable. On the other hand they only talk about sl_2. –  Noah Snyder Oct 14 '09 at 19:35

The formula is hard to implement but not difficult conceptually. The R-matrix formalism gives a link invariant for each irreducible representation. This is then extended to linear combinations of representations. For a 3-manifold invariant you take the regular representation. This means sum over irreducible representations the irreducible with scalar factor the quantum dimension. Then up to normalisation, this is a 3-manifold invariant by Kirby calculus.

-
Right, but the point of this question was to find somewhere these normalizations are explained in a lucid but precise way. –  Ben Webster Mar 7 '10 at 20:24