Volume conjecture

Volume conjecture

Field	Knot theory
Conjectured by	Hitoshi Murakami Jun Murakami Rinat Kashaev
Known cases	figure-eight knot three-twist knot Borromean rings torus knots all knots and links with volume zero twisted Whitehead links Whitehead doubles of nontrivial torus knots ${\displaystyle T(p,q)}$ with ${\displaystyle q=2}$
Consequences	Vassiliev invariants detect the unknot

In the branch of mathematics called knot theory, the volume conjecture is an open problem that relates quantum invariants of knots to the hyperbolic geometry of their complements.

Statement

Let O denote the unknot. For any knot ${\displaystyle K}$, let ${\displaystyle \langle K\rangle _{N}}$ be the Kashaev invariant of ${\displaystyle K}$, which may be defined as

${\displaystyle \langle K\rangle _{N}=\lim _{q\to e^{2\pi i/N}}{\frac {J_{K,N}(q)}{J_{O,N}(q)}}}$,

where ${\displaystyle J_{K,N}(q)}$ is the ${\displaystyle N}$-Colored Jones polynomial of ${\displaystyle K}$. The volume conjecture states that ^[1]

${\displaystyle \lim _{N\to \infty }{\frac {2\pi \log |\langle K\rangle _{N}|}{N}}=\operatorname {vol} (S^{3}\backslash K)}$,

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

${\displaystyle S^{3}\backslash K=\left(\bigsqcup _{i}H_{i}\right)\sqcup \left(\bigsqcup _{j}E_{j}\right)}$

with ${\displaystyle H_{i}}$ hyperbolic and ${\displaystyle E_{j}}$ Seifert-fibered. The simplicial volume${\displaystyle \operatorname {vol} (S^{3}\backslash K)}$ is then defined as the sum

${\displaystyle \operatorname {vol} (S^{3}\backslash K)=\sum _{i}\operatorname {vol} (H_{i})}$,

where ${\displaystyle \operatorname {vol} (H_{i})}$ is the hyperbolic volume of the hyperbolic manifold ${\displaystyle H_{i}}$. ^[1]

As a special case, if ${\displaystyle K}$ is a hyperbolic knot, then the JSJ decomposition simply reads ${\displaystyle S^{3}\backslash K=H_{1}}$, and by definition the simplicial volume ${\displaystyle \operatorname {vol} (S^{3}\backslash K)}$ agrees with the hyperbolic volume ${\displaystyle \operatorname {vol} (H_{1})}$.

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. ^{[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 by replacing the variable ${\displaystyle q}$ with the root of unity ${\displaystyle e^{i\pi /N}}$. They used an R-matrix as the 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:

If all Vassiliev invariants 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 ${\displaystyle K}$ is trivial, then ${\displaystyle J_{K,N}(q)=1}$ for any ${\displaystyle N}$.

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 (Tobias Ekholm), ^[5]
• The three-twist knot (Rinat Kashaev and Yoshiyuki Yokota), ^[5]
• The Borromean rings (Stavros Garoufalidis and Thang Le), ^[5]
• Torus knots (Rinat Kashaev and Olav Tirkkonen), ^[5]
• All knots and links with volume zero (Roland van der Veen), ^[5]
• Twisted Whitehead links (Hao Zheng), ^[6]
• Whitehead doubles of nontrivial torus knots ${\displaystyle T(p,q)}$ with ${\displaystyle q=2}$ (Hao Zheng). ^[6]

Relation to Chern-Simons theory

Using complexification, Murakami et al. (2002) proved that for a hyperbolic knot ${\displaystyle K}$,

${\displaystyle \lim _{N\to \infty }{\frac {2\pi \log \langle K\rangle _{N}}{N}}=\operatorname {vol} (S^{3}\backslash K)+CS(S^{3}\backslash K)}$,

where ${\displaystyle CS}$ is the Chern–Simons invariant. They established a relationship between the complexified colored Jones polynomial and Chern–Simons theory.

References

Notes

1. Murakami 2010, p. 17.
2. Kashaev, R.M. (1994-12-28). "Quantum Dilogarithm as a 6j-Symbol". Modern Physics Letters A . 09 (40): 3757–3768. arXiv: hep-th/9411147 . Bibcode:1994MPLA....9.3757K. doi:10.1142/S0217732394003610. ISSN   0217-7323.
3. Kashaev, R.M. (1995-06-21). "A Link Invariant from Quantum Dilogarithm". Modern Physics Letters A. 10 (19): 1409–1418. arXiv: q-alg/9504020 . Bibcode:1995MPLA...10.1409K. doi:10.1142/S0217732395001526. ISSN   0217-7323.
4. Kashaev, R. M. (1997). "The Hyperbolic Volume of Knots from the Quantum Dilogarithm". Letters in Mathematical Physics . 39 (3): 269–275. arXiv: q-alg/9601025 . Bibcode:1997LMaPh..39..269K. doi:10.1023/A:1007364912784.
5. Murakami 2010, p. 22.
6. Zheng, Hao (2007), "Proof of the volume conjecture for Whitehead doubles of a family of torus knots", Chinese Annals of Mathematics, Series B : 375–388, arXiv: math/0508138

Sources

• Murakami, Hitoshi (2010). "An Introduction to the Volume Conjecture". arXiv: 1002.0126 [math.GT]..
• 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 , Bibcode:1997LMaPh..39..269K, doi:10.1023/A:1007364912784 .
• Murakami, Hitoshi; Murakami, Jun (2001), "The colored Jones polynomials and the simplicial volume of a knot", Acta Mathematica , 186 (1): 85–104, arXiv: math/9905075 , doi: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 , 11 (1): 427–435, arXiv: math/0203119 , doi:10.1080/10586458.2002.10504485 .
• Gukov, Sergei (2005), "Three-Dimensional Quantum Gravity, Chern-Simons Theory, And The A-Polynomial", Communications in Mathematical Physics , 255 (1): 557–629, arXiv: hep-th/0306165 , Bibcode:2005CMaPh.255..577G, doi:10.1007/s00220-005-1312-y .