# Volume conjecture

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In the branch of [mathematics](/source/Mathematics) called [knot theory](/source/Knot_theory), the **volume conjecture** is an [open problem](/source/Open_problem) that relates [quantum invariants](/source/Quantum_invariant) of knots to the [hyperbolic geometry](/source/Hyperbolic_geometry) of their [complements](/source/Knot_complement).

## Statement

Let *O* denote the [unknot](/source/Unknot). For any knot K, let \langle K \rangle_N be the **Kashaev invariant** of K, which may be defined as

- \langle K \rangle_N=\lim_{q\to e^{2\pi i/N}}\frac{J_{K,N}(q)}{J_{O,N}(q)},

where J_{K,N}(q) is the N-[Colored Jones polynomial](/source/Jones_polynomial#Colored_Jones_polynomial) of K. The volume conjecture states that[1]

- \lim_{N\to\infty} \frac{2\pi\log |\langle K \rangle_N|}{N} = \operatorname{vol}(S^3 \backslash K),

where \operatorname{vol}(S^3 \backslash K) is the [simplicial volume](/source/Simplicial_volume) of the complement of K in the [3-sphere](/source/3-sphere), defined as follows. By the [JSJ decomposition](/source/JSJ_decomposition), the complement S^3 \backslash K may be uniquely decomposed into a system of tori

- S^3 \backslash K = \left( \bigsqcup_i H_i \right) \sqcup \left( \bigsqcup_j E_j \right)

with H_i [hyperbolic](/source/Hyperbolic_3-manifold) and E_j [Seifert-fibered](/source/Seifert-fibered). The **simplicial volume** \operatorname{vol}(S^3 \backslash K) is then defined as the sum

- \operatorname{vol}(S^3 \backslash K) = \sum_i \operatorname{vol}(H_i),

where \operatorname{vol}(H_i) is the [hyperbolic volume](/source/Hyperbolic_volume) of the hyperbolic manifold H_i.[1]

As a special case, if K is a [hyperbolic knot](/source/Hyperbolic_link), then the JSJ decomposition simply reads S^3 \backslash K = H_1, and by definition the simplicial volume \operatorname{vol}(S^3 \backslash K) agrees with the hyperbolic volume \operatorname{vol}(H_1).

## History

The Kashaev invariant was first introduced by Rinat M. Kashaev in 1994 and 1995 for hyperbolic links as a state sum using the theory of [quantum dilogarithms](/source/Quantum_dilogarithm).[2][3] Kashaev stated the formula of the volume conjecture in the case of hyperbolic knots in 1997.[4]

Murakami & Murakami (2001) pointed out that the Kashaev invariant is related to the [colored Jones polynomial](/source/Jones_polynomial#Colored_Jones_polynomial) by replacing the variable q with the root of unity e^{i\pi/N}. They used an [R-matrix](/source/R-matrix) as the [discrete Fourier transform](/source/Discrete_Fourier_transform) for the equivalence of these two descriptions. This paper was the first to state the volume conjecture in its modern form using the simplicial volume. They also prove that the volume conjecture implies the following conjecture of [Victor Vasiliev](/source/Victor_Vasiliev):

- If all [Vassiliev invariants](/source/Vassiliev_invariant) of a knot agree with those of the unknot, then the knot is the unknot.

The key observation in their proof is that if every Vassiliev invariant of a knot K is trivial, then J_{K,N}(q) = 1 for any N.

## Status

The volume conjecture is open for general knots, and it is known to be false for arbitrary links. The volume conjecture has been verified in many special cases, including:

- The [figure-eight knot](/source/Figure-eight_knot_(mathematics)) (Tobias Ekholm),[5]
- The [three-twist knot](/source/Three-twist_knot) (Rinat Kashaev and Yoshiyuki Yokota),[5]
- The [Borromean rings](/source/Borromean_ring) ([Stavros Garoufalidis](/source/Stavros_Garoufalidis) and [Thang Le](/source/Thang_Le)),[5]
- [Torus knots](/source/Torus_knot) (Rinat Kashaev and Olav Tirkkonen),[5]
- All knots and links with volume zero ([Roland van der Veen](/source/Roland_van_der_Veen)),[5]
- Twisted Whitehead links ([Hao Zheng](/source/Hao_Zheng)),[6]
- [Whitehead doubles](/source/Whitehead_double) of nontrivial torus knots T(p,q) with q=2 (Hao Zheng).[6]

## Relation to Chern-Simons theory

Using complexification, Murakami et al. (2002) conjectured that for a hyperbolic knot K,

- \lim_{N\to\infty} \frac{2\pi\log \langle K\rangle_N}{N} = \operatorname{vol}(S^3\backslash K) + CS(S^3\backslash K),

where CS is the [Chern–Simons invariant](/source/Chern%E2%80%93Simons_theory) of the frame field of the hyperbolic structure of K. This suggests a relationship between the colored Jones polynomial and complexified Chern–Simons theory.

## References

### Notes

1. Murakami 2010, p. 17.

1. Kashaev, R.M. (1994-12-28). ["Quantum Dilogarithm as a 6j-Symbol"](https://www.worldscientific.com/doi/abs/10.1142/S0217732394003610). *[Modern Physics Letters A](/source/Modern_Physics_Letters_A)*. **09** (40): 3757–3768. [arXiv:hep-th/9411147](https://arxiv.org/abs/hep-th/9411147). [Bibcode:1994MPLA....9.3757K](https://ui.adsabs.harvard.edu/abs/1994MPLA....9.3757K). [doi:10.1142/S0217732394003610](https://doi.org/10.1142/S0217732394003610). [ISSN 0217-7323](https://www.worldcat.org/issn/0217-7323)

1. Kashaev, R.M. (1995-06-21). ["A Link Invariant from Quantum Dilogarithm"](https://www.worldscientific.com/doi/abs/10.1142/S0217732395001526). *Modern Physics Letters A*. **10** (19): 1409–1418. [arXiv:q-alg/9504020](https://arxiv.org/abs/q-alg/9504020). [Bibcode:1995MPLA...10.1409K](https://ui.adsabs.harvard.edu/abs/1995MPLA...10.1409K). [doi:10.1142/S0217732395001526](https://doi.org/10.1142/S0217732395001526). [ISSN 0217-7323](https://www.worldcat.org/issn/0217-7323)

1. Kashaev, R. M. (1997). ["The Hyperbolic Volume of Knots from the Quantum Dilogarithm"](http://link.springer.com/10.1023/A:1007364912784). *[Letters in Mathematical Physics](/source/Letters_in_Mathematical_Physics)*. **39** (3): 269–275. [arXiv:q-alg/9601025](https://arxiv.org/abs/q-alg/9601025). [Bibcode:1997LMaPh..39..269K](https://ui.adsabs.harvard.edu/abs/1997LMaPh..39..269K). [doi:10.1023/A:1007364912784](https://doi.org/10.1023/A:1007364912784)

1. Murakami 2010, p. 22.

1. Zheng, Hao (2007), "Proof of the volume conjecture for Whitehead doubles of a family of torus knots", *[Chinese Annals of Mathematics, Series B](/source/Chinese_Annals_of_Mathematics,_Series_B)*. **28** (4): 375–388, [arXiv:math/0508138](https://arxiv.org/abs/math/0508138). [doi:10.1007/s11401-006-0373-3](https://doi.org/10.1007/s11401-006-0373-3)

### Sources

- Murakami, Hitoshi (2010). "An Introduction to the Volume Conjecture". [arXiv:1002.0126](https://arxiv.org/abs/1002.0126).
- Kashaev, Rinat M. (1997), "The hyperbolic volume of knots from the quantum dilogarithm", *Letters in Mathematical Physics*. **39** (3): 269–275, [arXiv:q-alg/9601025](https://arxiv.org/abs/q-alg/9601025). [Bibcode:1997LMaPh..39..269K](https://ui.adsabs.harvard.edu/abs/1997LMaPh..39..269K). [doi:10.1023/A:1007364912784](https://doi.org/10.1023/A:1007364912784).
- Murakami, Hitoshi & Murakami, Jun (2001), "The colored Jones polynomials and the simplicial volume of a knot", *[Acta Mathematica](/source/Acta_Mathematica)*. **186** (1): 85–104, [arXiv:math/9905075](https://arxiv.org/abs/math/9905075). [doi:10.1007/BF02392716](https://doi.org/10.1007/BF02392716).
- Murakami, Hitoshi; Murakami, Jun; Okamoto, Miyuki; Takata, Toshie; Yokota, Yoshiyuki (2002), "Kashaev's conjecture and the Chern-Simons invariants of knots and links", *[Experimental Mathematics](/source/Experimental_Mathematics_(journal))*. **11** (1): 427–435, [arXiv:math/0203119](https://arxiv.org/abs/math/0203119). [doi:10.1080/10586458.2002.10504485](https://doi.org/10.1080/10586458.2002.10504485).
- Gukov, Sergei (2005), "Three-Dimensional Quantum Gravity, Chern-Simons Theory, And The A-Polynomial", *[Communications in Mathematical Physics](/source/Communications_in_Mathematical_Physics)*. **255** (1): 557–629, [arXiv:hep-th/0306165](https://arxiv.org/abs/hep-th/0306165). [Bibcode:2005CMaPh.255..577G](https://ui.adsabs.harvard.edu/abs/2005CMaPh.255..577G). [doi:10.1007/s00220-005-1312-y](https://doi.org/10.1007/s00220-005-1312-y).

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