Samiksha Jaiswal (Editor)

Volume conjecture

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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.

Contents

Let O denote the unknot. For any knot K let ⟨ K ⟩ N be Kashaev's invariant of K ; this invariant coincides with the following evaluation of the N -colored Jones polynomial J K , N ( q ) of K :

Then the volume conjecture states that

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

Kashaev's Observation

Kashaev (1997) observed that the asymptotic behavior of a certain state sum of knots gives the hyperbolic volume 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 π 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 ⁡ i π N . They used R-matrix as the discrete fourier transformation for the equivalence of these two values.

The volume conjecture is important for knot theory. In the 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 C S ( S 3 ∖ K ) is called Chern-Simons invariant. They showed that there is a clear relation between the complexified colored Jones polynomial and Chern-Simons theory from mathematical point of view.

References

Volume conjecture Wikipedia
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