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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
##### Keywords:
link; hyperbolic volume; invariant; cyclic quantum dilogarithm
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