Chen and Yang have a more general version of the volume conjecture that they state for *all* hyperbolic $3$-manifolds (Conjecture 1.1 of [2]) including those with boundary. To do this, they have to define (citing Benedetti and Petronio [1]) a version of the Turaev-Viro invariant that assigns a manifold with boundary a number instead of a vector space.


My understanding is that they do this by triangulating the manifold and its boundary, and then dropping all the singular vertices that correspond to lower-dimensional strata and taking the usual state-sum over the remaining triangulation.
I think this means defining
$$
\mathrm{TV}\_r(M, q) = \mathrm{TV}\_r(M \setminus \partial M, q).
$$
If that's the case, is there a way to fit these numerical invariants into a TQFT-like structure?


One idea I had was as follows: If we have a link complement $M \setminus L$, the obvious thing to do would be to think of it as an embedded link $L \to M$, and I know how to think of an embedded link in a closed manifold as giving a number. In particular, $\mathrm{TV}\_r(L \to S^3)$ would agree with the norm-square of the Jones polynomial of $L$ evaluated at an appropriate root of unity. However, that's not what's going on in the construction, because the invariant Chen-Yang are considering takes different values on link complements.


[1] *Benedetti, Riccardo; Petronio, Carlo*, [**On Roberts’ proof of the Turaev-Walker theorem**](http://dx.doi.org/10.1142/S0218216596000266), J. Knot Theory Ramifications 5, No. 4, 427-439 (1996). [ZBL0890.57029](https://zbmath.org/?q=an:0890.57029).


[2] *Chen, Qingtao; Yang, Tian*, [**Volume conjectures for the Reshetikhin-Turaev and the Turaev-Viro invariants**](http://dx.doi.org/10.4171/QT/111), Quantum Topol. 9, No. 3, 419-460 (2018). [ZBL1405.57020](https://zbmath.org/?q=an:1405.57020), [arXiv:1503.02547](https://arxiv.org/abs/1503.02547).



How does the scalar TV invariant of a 3-manifold with boundary fit into the TQFT picture?
Based on the discussion in the comments with Ian Agol, here's a draft answer. I would welcome corrections/confirmation from anyone who knows more.


Let $M$ be an orientable manifold with possibly nonempty boundary, viewed as a cobordism $\emptyset \to \partial M $.
Then its $r$th Reshetikhin-Turaev invariant $\mathrm{RT}\_r(M)$ is a vector in $\mathrm{RT}\_r(\partial M)$, a vector space.
The TQFT axioms say that we can regard $\mathrm{RT}\_r(\overline M)$ as an element of the dual space $\mathrm{RT}\_r(\partial M)^\*$, where $\overline M$ is $M$ with opposite orientation.


We can pair the vector and covector to obtain an invariant
$$
\mathrm{TV}\_r(M) := \left \langle \mathrm{RT}\_r(\overline M), \mathrm{RT}\_r(M) \right \rangle \in \mathbb C
$$
which is in $\mathbb C$ even when $\partial M \ne \emptyset$. (Actually, I think it's always in $[0, \infty)$, and should be nonzero for any nontrivial $M$.) I believe that this is what Chen-Yang call the Turaev-Viro invariant of a manifold with boundary. This is closely related to the results in [arXiv:1701.07818](https://arxiv.org/abs/1701.07818), which discusses this construction for knot complements.


The idea is that, while $\mathrm{RT}\_r(\partial M)$ isn't quite a Hilbert space, there's at least an inner product on vectors coming from cobordisms, and we can exploit this to define $\mathrm{TV}\_r(M)$ as the norm of the vector $\mathrm{RT}\_r(M)$.


However, there now seems to be little relation between $\mathrm{TV}\_r(S^3 \setminus L)$ and the norm-square of the $r$th colored Jones polynomial of $L$ evaluated at a $r$th root of unity; as per [arXiv:1701.07818](https://arxiv.org/abs/1701.07818), the former involves a sum over the *lower-order* colored Jones polynomials at a different root of unity. Understanding this relationship better would be helpful in comparing the volume conjectures of Chen-Yang and of Kashaev-Murakami-Murakami.


