Then the volume conjecture states that
Rinat Kashaev (1997) observed that the asymptotic behavior of a certain state sum of knots gives the hyperbolic volume of the complement of knots and showed that it is true for the knots , , and . He conjectured that for general hyperbolic knots the formula (2) would hold. His invariant for a knot is based on the theory of quantum dilogarithms at the -th root of unity, .
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, . 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
where 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.
- 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.