Contents

# Contents

## Idea

### For knot complements

The original volume conjecture states that a certain limit of the colored Jones polynomial of a knot $K$ gives the simplicial volume of its complement in the 3-sphere.

$lim_{N \to \infty}\left(\frac{2 \pi log |V_N(K; q = e^{\frac{2 \pi i}{N}})|}{N}\right) = vol(K).$

Here $V_N(K; q)$ is the ratio of the values of the $N$-colored Jones polynomial of $K$ and of the unknot

$V_N(K; q) = \frac{J_N(K; q)}{J_N(\bigcirc; q)}.$

The simplicial volume of a knot complement can be found via its unique torus decomposition into hyperbolic pieces and Seifert fibered pieces by a system of tori. The simplicial volume is then the sum of the hyperbolic volumes of the hyperbolic pieces of the decomposition.

### Generally

More generally, volume conjectures state convergence of something like the Chern-Simons theory quantum invariant (realized notably via Reshetikhin-Turaev construction or Turaev-Viro construction) on hyperbolic 3-manifolds to the complex volume (for Reshetikhin-Turaev) or to the plain volume (for Turaev-Viro).

See (Chen-Yang 15)

## References

Original articles include

• Tudor Dimofte, Sergei Gukov, Quantum Field Theory and the Volume Conjecture, (arxiv:1003.4808)

• R. M. Kashaev, The Hyperbolic Volume Of Knots From The Quantum Dilogarithm Lett. Math. Phys. 39 (1997) 269-275.

• H. Murakami and J. Murakami, The Colored Jones Polynomial And The Simplicial Volume Of A Knot, Acta Math. 186 (2001) 85-104.

• H. Murakami, J. Murakami, M. Okamoto, T. Takata, and Y. Yokota, Kashaev’s Conjecture And The Chern-Simons Invariants Of Knots And Links, Experiment. Math. 11 (2002) 427-435.

• R. M. Kashaev, O. Tirkkonen, Proof of the volume conjecture for torus knots (arXiv:math/9912210)

• Sergei Gukov, Three-Dimensional Quantum Gravity, Chern-Simons Theory, And The A-Polynomial, Commun. Math. Phys. 255 (2005) 577-627.

• H. Murakami, Asymptotic Behaviors Of The Colored Jones Polynomials Of A Torus Knot, Internat. J. Math. 15 (2004) 547-555.

Generalization to Reshetikhin-Turaev construction on closed manifold, to the Turaev-Viro construction on manifolds with boundary, and to more general roots of unity than considered before is in

Review includes

• Edward Witten, pp. 4 of Two Lectures On The Jones Polynomial And Khovanov Homology (arXiv:1401.6996)

• Wikipedia, Volume conjecture

• H. Murakami, Asymptotic Behaviors Of The Colored Jones Polynomials Of A Torus Knot, Internat. J. Math. 15 (2004) 547-555.

Discussion in string theory includes

A conceptual explanation of the volume conjecture was proposed in

(but it seems that as a sketch or strategy for a rigorous proof, it didn’t catch on).

Last revised on September 12, 2018 at 10:14:40. See the history of this page for a list of all contributions to it.