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The hyperbolic volume of knots from the quantum dilogarithm. (English) Zbl 0876.57007
In the previous articles for the link \(L\) in the sphere \(S^3\) the author defined a link invariant \(\langle L\rangle\) depending on a positive integer \(N\) via three-dimensional interpretation of the cyclic quantum dilogarithm. This construction can be considered as an example of the three-dimensional topological quantum field theory (TQFT). In the article under review it is argued that this invariant is a quantum generalization of the hyperbolic volume invariant \(V(L)\) (the volume of the complement of the link \(L\) in \(S^3\) with a hyperbolic metric). Namely, studying some particular examples, the author shows for the value \(\langle L\rangle\) of this invariant on a link \(L\) the following asymptotic: \(2\pi\log|\langle L\rangle|\approx N V(L)\) as \(N\to\infty\).

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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