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 knot K let \( \langle K\rangle _{N} \) be Kashaev's invariant of K; this invariant coincides with the following evaluation of the N {\displaystyle N} N-colored Jones polynomial \( J_{{K,N}}(q) \_ of K:
\( \langle K\rangle _{N}=\lim _{{q\to e^{{2\pi i/N}}}}{\frac {J_{{K,N}}(q)}{J_{{O,N}}(q)}}. \) (1)
Then the volume conjecture states that
\( \lim _{{N\to \infty }}{\frac {2\pi \log |\langle K\rangle _{N}|}{N}}=\operatorname {vol}(K),\, \) (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 \( \operatorname {vol}(K) \) of the complement of knots K and showed that it is true for the knots \( 4_{1} \) , \( 5_{2} \), and \( 6_{1} \). He conjectured that for the general hyperbolic knots the formula (2) would hold. His invariant for a knot K is based on the theory of quantum dilogarithms at the N-th root of unity, \( q=\exp {(2\pi i/N)} \).
Colored Jones Invariant
Murakami & Murakami (2001) had firstly pointed out that Kashaev's invariant is related to Jones polynomial by replacing q with the 2N-root of unity, namely, \( \exp {{\frac {i\pi }{N}}} \) . 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
\( \lim _{{N\to \infty }}{\frac {2\pi \log \langle K\rangle _{N}}{N}}=\operatorname {vol}(S^{3}\backslash K)+CS(S^{3}\backslash K), \) (3)
where \( 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.
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