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A. We study quantum invariant Z γ (M) for cusped hyperbolic 3-manifold M. We construct this invariant based on oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in studies of the modular double of the quantum group. Following Thurston and Neumann-Zagier, we deform a complete hyperbolic structure of M, and correspondingly we define quantum invariant Z γ (M u ). This quantum invariant is shown to give the Neumann-Zagier potential function in the classical limit γ → 0, and the Apolynomial can be derived from the potential function. We explain our construction by taking examples of 3-manifolds such as complements of hyperbolic knots and punctured torus bundle over the circle. ISince the quantum invariant of knots/links and 3-manifolds as a generalization of the Jones polynomial [25] is constructed by Witten [53] by use of the Chern-Simons path integral, studies on quantum invariants have been much developed. Recently geometrical interpretations of the quantum invariants have received interests since Kashaev observed an intriguing relationship [29] between the hyperbolic volume and his knot invariant, which is later identified with a specific value of the N-colored Jones polynomial J K N; e 2πi/N [41] (here the N-colored Jones polynomial is normalized to be J unknot (N; q) = 1). Namely the hyperbolic volume of the knot complement S 3 \ K is conjectured to dominate the asymptotics of the invariant J K N; e 2πi/N in the large-N limit N → ∞,

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The extended Bloch groups of biquadratic and dihedral number fields

Guo

Qin

2018

Journal of Pure and Applied Algebra

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Cited by 7 publications

References 18 publications

Three Questions about Simplices in Spherical and Hyperbolic 3-Space

Three Questions about Simplices in Spherical and Hyperbolic 3-Space

Generalized volume conjecture and the A-polynomials: The Neumann–Zagier potential function as a classical limit of the partition function

The extended Bloch groups of biquadratic and dihedral number fields

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