Volume conjecture

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

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

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 R. 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 Vassiliev:

Every knot that is different from the trivial knot has at least one different Vassiliev (finite type) invariant.

Status

The volume conjecture is open in the general case, but it has been verified for many special cases, including^[1]

• the figure-eight knot (Ekholm),
• the three-twist knot (Kashaev and Yokota),
• Whitehead doubles of torus knots (Zheng),
• torus knots (Kashaev and Tirkkonen),
• knots and links with volume zero (van der Veen).

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

• 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", Commun. Math. Phys., 255 (1): 557–629, arXiv:hep-th/0306165, Bibcode:2005CMaPh.255..577G, doi:10.1007/s00220-005-1312-y.