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**, J. Knot Theory Ramifications 5, No. 4, 427-439 (1996). ZBL0890.57029.

[2] *Chen, Qingtao; Yang, Tian*, **Volume conjectures for the Reshetikhin-Turaev and the Turaev-Viro invariants**, Quantum Topol. 9, No. 3, 419-460 (2018). ZBL1405.57020, arXiv:1503.02547.

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