Attached to any object $V\in \mathcal{C}$ of a ribbon category $\mathcal{C}$, Reshetikhin and Turaev have defined knot invariants $$\tau_V(K)\ \in\ \text{End}_{\mathcal{C}}(1_{\mathcal{C}})$$ for instance see Turaev's "Quantum Invariants of Knots and 3-Manifolds". In examples this is often just a complex number or element of some ring. The argument is something like
- There is (?) a unique functor $F:\text{Rib}\to \mathcal{C}$ of ribbon categories, where $\text{Rib}$ is the category of oriented 0-manifolds with morphisms the framed cobordisms between them, such that $F(+)=V$.
(As stated in the reference we need to use a bigger category than $\text{Rib}$ including coupons etc., but that's not needed for knot invariants and so one might hope that the simplified version I wrote above is true, if it's not true then in my question just replace $\text{Rib}$ with that larger category). Then,
- View a (framed) knot $K\subseteq S^3$ as a (framed) cobordism $K:\varnothing\to \varnothing$, then we get $F(K):1_\mathcal{C}\to 1_\mathcal{C}$, which is our invariant.
My question is then: say there is a fully extended 3d TQFT $\mathcal{T}$ such that $$\mathcal{T}(S^1)\ =\ \mathcal{C}.$$ Does the functor $F:\text{Rib}\to \mathcal{C}$, hence the knot invariant, have a simple description in terms of $\mathcal{T}$? I assume the answer is yes, but I haven't been able to come up with it.
