# Why is the volume conjecture important?

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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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. –  Ryan Budney Sep 1 '10 at 4:13
A blog post on this question: ldtopology.wordpress.com/2011/11/11/… –  Daniel Moskovich Nov 11 '11 at 15:03

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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