TITLE: Is there a volume conjecture for closed 3-manifolds?
QUESTION [19 upvotes]: A typical statement of the volume conjecture, for instance in Murakami's survey 1002.0126, is

Conjecture: For $K$ a knot in $S^3$, the N-th colored Jones polynomials are related to the volume of the knot complement by $$ 2 \pi \lim_{N \rightarrow \infty} \frac{1}{N} \log | J_N(K; \exp(2\pi i / N)) | = Vol( S^3 \backslash K).$$

A refinement of the conjecture is $ 2 \pi \lim_{N \rightarrow \infty} \frac{1}{N} \log J_N(K; \exp(2\pi i / N)) = Vol( S^3 \backslash K) + i \; CS( S^3 \backslash K) \; (\mod \pi^2 i)$
where CS is the Chern-Simons invariant.  Both sides of the conjecture can be formulated for 3-manifolds more general than knots in S^3 and their complements.  In particular, we might ask about closed 3-manifolds without a knot at all.

Question: Is there an analogous volume conjecture for (some) closed 3-manifolds, or for closed 3-manifolds with embedded knots, and if so where in the literature are these formulations discussed?

REPLY [6 votes]: There's another volume conjecture formulated by Chen and Yang for Turaev-Viro invariants of closed manifolds. They present some evidence for the conjecture in the paper. In a second paper, Yang and collaborators formulate another volume conjecture based on the colored Jones polynomial, but evaluated at different roots of unity: see Question 1.7. They prove this version for the figure eight knot.