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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
SnapPea
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