# Framings in the definition of Reshetikhin-Turaev TQFT

I posted the following question at Mathe Stack Exchange.link text But it has not yet answered. I am sorry if you check both sites but I also want people here to look at this problem.

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and then they extend it to a TQFT.

First, let me briefly describe how to define this TQFT in the followings.

Let $(M, \partial_{-}M, \partial_{+}M)$ be a cobordism. Let $\Omega$ be a ribbon graph in $M$. To define a TQFT, we first glue standard handlebodies with standard ribbon graphs $R$ (defined below) inside to the bottom boundary $\partial_{-}M$ by a given parametrization and also glue them to the top boundary $\partial_{+}M$ by a composition of a given parametrization and reflection map.

Then we get a closed 3 manifold with a ribbon graph $\Omega'$, which is obtained by gluing $\Omega$ and $R$. We apply the invariant $\tau$ to this closed 3-manifold to obtain a TQFT.

My question is that when we glue standard handlebodies to the boundaries, how do we define a framing of ribbon graphs, which are images of $R$. We need to know framing to calculate $\tau$.

$R$ consistes of a coupon (a rectangle) and $g$ cap like bands attaching the coupon and several bands attaching one end to this coupon and the other end attached to the boundary of the handlebody. Here $g$ is a genus of the handlebody.

If a hundlebody is genus $1$, then I think the framing can be determined by the image of meridian. But if a genus is greater than $1$, I don't know how to define a framing.

The book and the paper don't mention how to define framings.

Any help is apprecited. Thank you in advance

• Each open edge is framed by the linking number of the ribbon with its core (equivalently: a trivialization of normal bundle of the core). The compatibility condition is that the framings coincide at vertices- the three ribbons glue together, with matching orientations, to form a disc (the coupon). – Daniel Moskovich Jun 1 '12 at 10:27
• I don't understand your question. Are you talking about framings of the ribbon graph, or framings of the 3-manifold? A ribbon graph is equivalent to a framed graph, so if you are starting with a ribbon structure on the graph(s) there's no need to say anything additional about framings. – Kevin Walker Jun 1 '12 at 13:12
• @Kevin Walker, I mean a framing of a ribbon graph. The standard ribbons in the standard handlebody has framing $0$, by definition. I want to know the framing of the ribbon graph after it is glued via parametrizations to cobordism to form a closed manifold. – Primo Jun 2 '12 at 8:56
• @Daniel Moskovich, I don't understand well. Could you give me examples that illustrate your comment? – Primo Jun 2 '12 at 13:27

A framing of a graph inside a 3-manifold $M$ is, by definition, a choice of (isotopy class of) ribbon whose core is the original graph. If the graph is a single loop and if $M$ is an integer homology sphere, then we can identity framings with integers (the linking number of a boundary component of the ribbon with its core). Otherwise we can't. So saying that the "standard ribbons in the standard handlebody has framing 0" is incorrect -- you can't describe a framing in a handlebody with an integer (such as 0).
If we glue two handlebodies together to form a closed 3-manifold $M$, and if each handlebody contains a ribbon graph, then image under gluing of the union of these two ribbon graphs is a ribbon graph in $M$, i.e. a framed graph in $M$.