Volume conjecture

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.