Kashaev's invariant and the volume of a hyperbolic knot after Y. Yokota (Q2751978)

This is an expository paper to describe Yokota's method explaining the beautiful connection between Kashaev's invariants of hyperbolic knots and links and the hyperbolic structures on their complements, putting special emphasis on the figure-eight knot. NEWLINENEWLINENEWLINE\textit{R. M. Kashaev} [A link invariant from quantum dilogarithm, Mod. Phys. Lett. A 10, No. 19, 1409--1418 (1995; Zbl 1022.81574)] introduced a link invariant by using the quantum dilogarithm and, after some concrete calculations for the three simplest hyperbolic knots, he conjectured that his invariant determines the hyperbolic volume for hyperbolic links. \textit{J. Murakami} and the author proved [Acta Math. 186, No. 1, 85--104 (2001; Zbl 0983.57010), see above] that Kashaev's invariant coincides with a colored Jones polynomial evaluated at a root of unity. ``By using this, Y. Yokota, following D. Thurston, considered a polyhedron decomposition corresponding to the \(R\)-matrix used to define the colored Jones polynomials and showed that for at least one example, the knot \(6_2\), this decomposition gives a nice and constructive proof for Kashaev's conjecture (he also mentioned that this seems work well for other knots, especially for alternating knots). After that Yokota used Kashaev's original \(R\)-matrix and found another tetrahedron decomposition of a hyperbolic knot complement fit for the asymptotic behavior of Kashaev's invariant. This gives simpler decompositions and seems to work much better including non-alternating knots''.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00054].