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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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    $\begingroup$ Given a surface, one may consider the formal linear span of all the manifolds bounding it. arxiv.org/abs/math/0503054 One may pair two such manifolds together by gluing along the surface, then evaluate $TV_r$ on the closed manifold. Then $TV_r$ gives a pairing on this space, by bilinear extension. One can quotient by the null-space of this pairing, maybe this gives a finite-dimensional vector space associated to the surface? However, I think the pairing is indefinite in the Chen-Yang case. $\endgroup$ – Ian Agol Jun 23 '20 at 22:58
  • $\begingroup$ Applying your comment, a manifold with boundary would be a linear functional on the vector space associated to its boundary. I suppose this is also what happens in the usual formalism, but here the vector space has a much more topological interpretation than the usual combinatorial state space of TV theory. I'm still not sure how to get a number instead of a covector in a natural, topological way. Maybe you close up the manifold in some canonical fashion? $\endgroup$ – Calvin McPhail-Snyder Jun 24 '20 at 13:48
  • $\begingroup$ Actually, what I suggested probably doesn’t work, given that TV is the square of RT. sciencedirect.com/science/article/pii/0040938394000530 $\endgroup$ – Ian Agol Jun 24 '20 at 15:10
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    $\begingroup$ Maybe for a manifold with boundary, it’s the inner product of the appropriate RT invariant with itself (in the RT vector space associated to the boundary, which may have an indefinite inner Hermitian inner product)? $\endgroup$ – Ian Agol Jun 24 '20 at 15:12
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    $\begingroup$ I think arxiv.org/pdf/1807.03327.pdf Proposition 5.3. may answer your question. $\endgroup$ – Tian Yang Aug 3 '20 at 21:37
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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, 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, 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.

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