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Generalized Kashaev invariants for knots in three manifolds. (English) Zbl 1365.57020
The aim of the paper is a generalization of Kashaev’s invariants for a knot in the 3-sphere to invariants of a knot in a 3-manifold. From a study of quantum dilogarithms, Kashaev introduced an invariant of links in 3-manifolds, gave an \(R\)-matrix formulation of his invariant for knots in the 3-sphere and found a relation between his invariant and the hyperbolic volume of knot complements in \(S^3\), resulting in a conjecture not yet rigorously proved. A complexified refinement of Kashaev’s conjecture for hyperbolic knots in the 3-sphere, in terms of the hyperbolic volume and the Chern-Simons invariant, has been given by H. Murakami et al. [Exp. Math. 11, No. 3, 427–435 (2002; Zbl 1117.57300)].
“The aim of the present paper is to construct certain quantum invariants for knots in 3-manifolds which have a relation to the hyperbolic volume as the above conjectures.” “We construct a family of invariants of a knot in a 3-manifold by combining the invariant of 3-manifolds by M. Hennings [J. Lond. Math. Soc., II. Ser. 54, No. 3, 594–624 (1996; Zbl 0882.57002)] and the logarithmic invariant of knots in \(S^3\) [J. Murakami and K. Nagamoto, Int. J. Math. 19, No. 10, 1203–1213 (2008; Zbl 1210.57016)]. This family contains a generalized Kashaev invariant which coincides with Kashaev’s invariant for the case of the 3-sphere.”
The author states a volume conjecture for his generalized Kashaev invariant, again in terms of the volume of a hyperbolic knot in a 3-manifold and the Chern-Simons invariant, and checks the conjecture numerically for some knots in lens spaces.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
16T05 Hopf algebras and their applications
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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