The volume conjecture, a formula relating hyperbolic volume of a knot complement with the semiclassical limit of a family of coloured Jones polynomials, is widely considered the biggest open problem in quantum topology. It is one of a large family of conjectures and research programmes which have to do with detecting classical geometry with semiclassical limits.
Embarassing as it is to say in public, I only very partially understand why people care so much about such conjectures.

What fantastic consequences would there be for low dimensional topology if the volume conjecture were proven tomorrow? What if all the related conjectures were proven too? How would it improve our understanding of classical topology? More broadly, how would it advance mathematics?
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    $\begingroup$ A worst-case concern could be that quantum invariants of knots are a type of one-way function in the sense of crytography. They are readily computable but it might be extremely difficult to compute strong topological data from them, like say the symmetry group of a knot, or the Gromov norm of a link complement. I see the persistence of the volume conjecture as a suggestion this might not be the case. $\endgroup$ – Ryan Budney Sep 1 '10 at 4:13
  • $\begingroup$ A blog post on this question: ldtopology.wordpress.com/2011/11/11/… $\endgroup$ – Daniel Moskovich Nov 11 '11 at 15:03

I don't know any consequences for low-dimensional topology. My impression is that it would indicate how powerful TQFT invariants are at distinguishing 3-manifolds or knots. For example, the volume conjecture (or maybe variants by the Murakamis) would imply that the colored Jones polynomials distinguish knots from the unknot. This corollary is also claimed by Jorgen Andersen. There are only finitely many hyperbolic knots of a given volume, so the volume conjecture would imply that the colored Jones polynomials are close to distinguishing hyperbolic knots.

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    $\begingroup$ In particular, if colored Jones polynomials distinguish the unknot then it follows that finite type invariants distinguish the unknot. $\endgroup$ – Noah Snyder Sep 1 '10 at 4:30
  • $\begingroup$ But it would NOT tell you which easy-to-calculate quantum invariant would distinguish between given manifold M and given manifold N. Playing devil's advocate, what use would proving the power of the set of ALL quantum invariants (let's toss in the coloured HOMFLYPTs as well) be if it doesn't help us to calculate anything concrete? Doesn't the fundamental group (and peripheral group, if you like) do that already? Also, as you point out, Andersen can prove that the coloured Jones distinguish the unknot, without resorting to anything nearly as heavy as the volume conjecture. $\endgroup$ – Daniel Moskovich Sep 1 '10 at 11:47
  • $\begingroup$ Regarding Andersen's claim: is there any text relating to this claim, such as a paper outlining a program, slides or notes of a relevant talk, or at least an abstract containing a written claim? I understand it's an old story (5 years? more?) but people still seem to be enthusiastic about it - apparently for a good reason? @Noah Snyder: is there any reference for this implication? $\endgroup$ – Sergey Melikhov Nov 29 '10 at 3:42
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    $\begingroup$ @ Sergey: I don't know of any paper, I've only seen him claim it in a colloquium talk and in conversations. He has a way of defining quantum invariants related to SU(2) representations (I think this is joint work with Ueno). He uses asymptotics of Toeplitz operators to show that these invariants detect non-trivial SU(2) reps. Then he claims that these invariants are equivalent to the Reshetikhin-Turaev invariants, which for Dehn surgery on a knot may be computed by the colored Jones polynomials. Finally, work of Kronheimer-Mrowka imply that non-trivial Dehn surgeries on knots have SU(2) reps. $\endgroup$ – Ian Agol Nov 29 '10 at 4:21

Someone else will have to discuss the applications in topology, but I can point out at least one reason the volume conjecture is interesting.

It's often said that no one knows how to define the functional integral for Chern-Simons theory. This isn't literally true. The Reshetikhin-Turaev construction can be interpreted -- tautologically -- as defining a volume measure on a certain space of functionals. (This is just like in quantum mechanics, where one interprets the kernel $\langle q_i|e^{-Ht}|q_f\rangle$ as the volume of the space of paths $\phi: [0,t] \to \mathbb{R}$ which begin at $q_i$ and end at $q_f$.) What we don't know how to do is define the path integral measure as a continuum limit of regularized integrals that look like $\frac{1}{Z}e^{iCS(A)}dA$.

The volume conjecture (in particular the version where log of the Jones polynomial looks like vol(3-manifold) plus i times the Chern-Simons functional) tells us that the tautological measure you get from Reshetikhin-Turaev actually has something to do with the Chern-Simons action!


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