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Kashaev’s conjecture and the Chern-Simons invariants of knots and links. (English) Zbl 1117.57300
Summary: R. M. Kashaev [Mod. Phys. Lett. A 10, 19, 1409–1418 (1995; Zbl 1022.81574)] conjectured that the asymptotic behavior of the link invariant he introduced, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots \(6_3\), \(8_9\) and \(8_{20}\) and for the Whitehead link, the colored Jones polynomials are related to the hyperbolic volumes and the Chern–Simons invariants and propose a complexification of Kashaev’s conjecture.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
33B30 Higher logarithm functions
57M50 General geometric structures on low-dimensional manifolds
58J28 Eta-invariants, Chern-Simons invariants
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
Software:
SnapPea
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