an:06718139
Zbl 1365.57020
Murakami, Jun
Generalized Kashaev invariants for knots in three manifolds
EN
Quantum Topol. 8, No. 1, 35-73 (2017).
00365333
2017
j
57M27 16T05 17B37 81R50
knot in a 3-manifold; hyperbolic knot; volume; Kashaev invariant; volume conjecture
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 \textit{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 \textit{M. Hennings} [J. Lond. Math. Soc., II. Ser. 54, No. 3, 594--624 (1996; Zbl 0882.57002)] and the logarithmic invariant of knots in \(S^3\) [\textit{J. Murakami} and \textit{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.
Bruno Zimmermann (Trieste)
Zbl 1117.57300; Zbl 0882.57002; Zbl 1210.57016