In the branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

called

knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...

, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of

knot complement In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...

s. Let ''O'' denote the

unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...

. 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

- Colored Jones Polynomial

J_(q)

K

: Then the volume conjecture states that where vol(''K'') denotes the

hyperbolic volume In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological inv ...

of the complement of ''K'' in the

3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...

Kashaev's Observation

observed that the asymptotic behavior of a certain state sum of knots gives the

\operatorname(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 knot Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

s the formula (2) would hold. His invariant for a knot

K

is based on the theory of

quantum dilogarithm In mathematics, the quantum dilogarithm is a special function defined by the formula : \phi(x)\equiv(x;q)_\infty=\prod_^\infty (1-xq^n),\quad , q, 0. References * * * * * * * External links * {{nlab, id=quantum+dilogarithm, title=quantum ...

s at the

N

-th root of unity,

q=\exp

Colored Jones Invariant

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,

\exp

. They used an

R-matrix The term R-matrix has several meanings, depending on the field of study. The term R-matrix is used in connection with the Yang–Baxter equation. This is an equation which was first introduced in the field of statistical mechanics, taking its n ...

as the

discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...

for the equivalence of these two values. The volume conjecture is important for

. In section 5 of this paper they state that: : Assuming the volume conjecture, every knot that is different from the

trivial knot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...

has at least one different Vassiliev (finite type) invariant.

Relation to Chern-Simons theory

Using complexification, rewrote the formula (1) into 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

*. *. *. *{{citation , last1 = Gukov , first1 = Sergei , doi = 10.1007/s00220-005-1312-y , arxiv = hep-th/0306165 , issue = 1 , journal = Commun. Math. Phys. , pages = 557-629 , title = Three-Dimensional Quantum Gravity,　Chern-Simons Theory, And　The A-Polynomial　 , volume = 255 , year = 2005, bibcode = 2005CMaPh.255..577G. Knot theory Conjectures Unsolved problems in geometry