# Is there a volume conjecture for closed 3-manifolds?

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?

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There is such a conjecture. See:
Optimistic calculations about the Witten--Reshetikhin--Turaev invariants of closed three-manifolds obtained from the figure-eight knot by integral Dehn surgeries. Hitoshi Murakami. Surikaisekikenkyusho Kokyuroku No. 1172, (2000), 70--79.
It roughly states that 2πi times the "optimistic limit" (sort-of defined by Murakami) as N goes to infinity of the quantum SU(2) invariant of 3-manifold M, of level N, divided by N, equals the Chern-Simons invariant of M plus i times its hyperbolic volume. Not all terms are rigorously defined for all closed 3-manifolds, and part of the conjecture seems to be that there exist rigourous definitions of all terms in this general setting. This conjecture, and other related conjectures, are discussed in Section 7 (particularly in Section 7.3) of
Problems on invariants of knots and 3–manifolds. Tomotada Ohtsuki. Geometry & Topology Monographs 4 (2002) 377–572.

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Daniel -- when I read this paper, I wasn't able to give a precise meaning to the "optimistic limit" for general knots. Hiroshi Murakami told me a couple of years ago that he can't do this either. (Otherwise, I would imagine, he would have put it in the paper.) So I am not sure this qualifies as a conjecture. –  algori Feb 25 '10 at 18:58
@algori Definitely! It's only a "conjecture" in a loose sense of the word. –  Daniel Moskovich Mar 1 '10 at 5:40

There are actually some hyperbolic volume results due to Francois Costantino which you can find on his web page

http://www-irma.u-strasbg.fr/~costanti/Papers%20and%20preprints.html

Especially the papers in the Proceedings of the London Math Society and Geometry and Topology.

Also Stavros has an interesting conjecture about the Reshitkhin-Turaev invariant of closed manifolds and the complex Chern-Simons invariant, which includes volume.

http://arxiv.org/abs/0711.1716

Stavros idea is to form a generating function whose coefficients are the Reshetikhin-Turaev invariants of the manifold of level $r$. He proves that the power series converges in a neighborhood of zero in the plane, and then conjectures that the Borel regulator of the manifold has something to do with the poles of the analytic continuation.

More speculatively, using the standard values of $q$, that is $e^{2\pi i/r}$ where $r\geq 3$ is an integer, the Reshetikhin-Turaev invariant of a three-manifold grows polynomially, where the exponent is half the complex dimension of its $SL(2,C)$-character variety, so you don't get exponential growth. This means the asymptotics of the values of the invariants are quite subtle.

On the other hand, by picking other primitive $r$-th roots of unity, the positivity of the quantum dimensions of the representations break down and you can get exponential growth. The failure of positivity is broached in Habegger, Masbaum, Vogel and Blanchet's big paper in Topology on TQFT. Anyways...

I don't know of the results of any experiments about the growth of the Reshetikhin-Turaev invariant with such choices. My guess is that there is interesting stuff going on with the exponential growth rate, that is reflecting the geometry of the underlying manifold.

In a slightly different direction, using "bad" values of $q$ as above, Gregor Masbaum, Jorgen Anderson and Kenji Ueno were able to recover the translation length of an element of the mapping class group of a planar surface with four boundary components on Teichm\"{u}ller space from the representation on the state space assigned to the surface by the TQFT with corners underlying the Reshetikhin-Turaev invariant. They get it as the exponential growth rate of the trace of the induced morphism.

See for instance:

Andersen, Jørgen Ellegaard; Masbaum, Gregor; Ueno, Kenji Topological quantum field theory and the Nielsen-Thurston classification of $M(0,4)$. Math. Proc. Cambridge Philos. Soc. 141 (2006), no. 3, 477--488

Not only has the volume conjecture not been resolved, it is just the tip of the iceberg when it comes to detecting classical geometry from semiclassical limits.

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At this point, is there some general heuristic or big-picture conceptual framework for all these (usually conjectural) asymptotic-behavior results? Some underlying (conjectural) explanation of what's going on, among experts? Or is that too much to ask for right now? –  Sam Lewallen Mar 7 '10 at 7:06
There are analytic methods for making predictions about the semiclassical limit, that give a heuristic explanation of what to expect. However in even the simplest concrete situtations the predicted answers turn out to be false.IMHO the two biggest obstructions are:1) The singularities of character varieties of three-manifold groups are not understood well enough, so that the method of stationary phase can be adapted to the situation rigorously. 2) The phenomenon depends on the breakdown of positivity, so there aren't well developed analytic methods –  Charlie Frohman Mar 7 '10 at 11:47

Check out Questions 2.6 and 2.9 in this paper of Constantino, Geer, and Patureau-Murand.

There are also physics papers considering versions of the volume conjecture, although I'm not sure if there is a precisely defined mathematical conjecture. See e.g. the papers of Hikami, or Dimofte and Gukov.

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