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 be Kashaev's invariant of ; this invariant coincides with the following evaluation of the -Colored Jones Polynomial of :
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(1)
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Then the volume conjecture states that
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(2)
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where vol(K) denotes the hyperbolic volume of the complement of K in the 3-sphere.
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, .
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:
Using complexification, Murakami et al. (2002) rewrote the formula (1) into
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(3)
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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.