The Kashaev and quantum hyperbolic link invariants

Stéphane Baseilhac (1) , Riccardo Benedetti (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 Dipartimento di matematica

We show that the link invariants derived from 3-dimensional quantum hyperbolic geometry can be defined via planar state sums based on link diagrams and a new family of enhanced Yang-Baxter operators (YBO) that we compute explicitly. By a local comparison of the respective YBO's we show that these invariants coincide with Kashaev's specializations of the colored Jones polynomials. As a further appli- cation we disprove a conjecture about the semi-classical limits of quantum hyperbolic invariants, by showing that it conflicts with the existence of hyperbolic links that verify the volume conjecture.

Mots clés: volume conjecture; Links; colored Jones polynomials; generalized Alexander invariants; quantum hyperbolic geometry; Yang-Baxter operators; volume conjecture.

Domaines: Topologie géométrique [math.GT]; Algèbres quantiques [math.QA]

Stéphane Baseilhac, Riccardo Benedetti. The Kashaev and quantum hyperbolic link invariants. Journal of Gökova Geometry Topology, 2011, 5, pp.31-85. ⟨hal-00768043⟩