nLab
volume conjecture

Contents

Contents

Idea

For knot complements

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

lim N(2πlog|V N(K;q=e 2πiN)|N)=vol(K). 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)V_N(K; q) is the ratio of the values of the NN-colored Jones polynomial of KK and of the unknot

V N(K;q)=J N(K;q)J N(;q). 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.