In the branch of mathematics called knot theory

Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...

, the volume conjecture is the following open problem that relates quantum invariant

Quantum invariant

In the mathematical field of knot theory, a quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.-List of invariants:*Finite type invariant*Kontsevich invariant*Kashaev's invariant...

s of knots to the hyperbolic geometry of knot complement

Knot complement

In mathematics, the knot complement of a tame knot K is the complement of the interior of the embedding of a solid torus into the 3-sphere. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a thickened neighborhood of K; so N is a solid torus...

s.

Let O denote the unknot

Unknot

The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...

. For any knot K let be Kashaev's invariant of ; this invariant coincides with the following evaluation of the -colored Jones polynomial  of :

Then the volume conjecture states that

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

3-sphere

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

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