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

Let O denote the unknot. For any hyperbolic knot K let ${\displaystyle \langle K\rangle _{N}}$ be Kashaev's invariant of ${\displaystyle K}$; this invariant coincides with the following evaluation of the ${\displaystyle N}$-Colored Jones Polynomial ${\displaystyle J_{K,N}(q)}$ of ${\displaystyle K}$:

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

(1)

Then the volume conjecture states that

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

(2)

where vol(K) denotes the hyperbolic volume of the complement of K in the 3-sphere.

Kashaev's Observation

Rinat Kashaev (1997) observed that the asymptotic behavior of a certain state sum of knots gives the hyperbolic volume ${\displaystyle \operatorname {vol} (K)}$ of the complement of knots ${\displaystyle K}$ and showed that it is true for the knots ${\displaystyle 4_{1}}$, ${\displaystyle 5_{2}}$, and ${\displaystyle 6_{1}}$. He conjectured that for general hyperbolic knots the formula (2) would hold. His invariant for a knot ${\displaystyle K}$ is based on the theory of quantum dilogarithms at the ${\displaystyle N}$-th root of unity, ${\displaystyle q=\exp {(2\pi i/N)}}$.

Colored Jones Invariant

Murakami & Murakami (2001) had firstly pointed out that Kashaev's invariant is related to the colored Jones polynomial by replacing q with the 2N-root of unity, namely, ${\displaystyle \exp {\frac {i\pi }{N}}}$. They used an R-matrix as the discrete Fourier transform for the equivalence of these two values.

The volume conjecture is important for knot theory. In section 5 of this paper they state that:

Assuming the volume conjecture, every knot that is different from the trivial knot has at least one different Vassiliev (finite type) invariant.

Relation to Chern-Simons theory

Using complexification, Murakami et al. (2002) rewrote the formula (1) into

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

(3)

where ${\displaystyle CS(S^{3}\backslash K)}$ is called the Chern–Simons invariant. They showed that there is a clear relation between the complexified colored Jones polynomial and Chern–Simons theory from a mathematical point of view.

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, 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.