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Article Dans Une Revue Algebraic and Geometric Topology Année : 2015

Relations between Witten-Reshetikhin-Turaev and non semi-simple sl(2) 3-manifold invariants

Résumé

The Witten-Reshetikhin-Turaev invariants extend the Jones polynomials of links in S^3 to invariants of links in 3-manifolds. Similarly, in a preceding paper, the authors constructed two 3-manifold invariants N_r and N^0_r which extend the Akutsu-Deguchi-Ohtsuki invariant of links in S^3 colored by complex numbers to links in arbitrary manifolds. All these invariants are based on representation theory of the quantum group Uqsl2, where the definition of the invariants N_r and N^0_r uses a non-standard category of Uqsl2-modules which is not semi-simple. In this paper we study the second invariant N^0_r and consider its relationship with the WRT invariants. In particular, we show that the ADO invariant of a knot in S^3 is a meromorphic function of its color and we provide a strong relation between its residues and the colored Jones polynomials of the knot. Then we conjecture a similar relation between N^0_r and a WRT invariant. We prove this conjecture when the 3-manifold M is not a rational homology sphere and when M is a rational homology sphere obtained by surgery on a knot in S^3 or when M is a connected sum of such manifolds.

Dates et versions

hal-00872107 , version 1 (11-10-2013)

Identifiants

Citer

Francesco Costantino, Nathan Geer, Bertrand Patureau-Mirand. Relations between Witten-Reshetikhin-Turaev and non semi-simple sl(2) 3-manifold invariants. Algebraic and Geometric Topology, 2015, pp.1363--1386. ⟨10.2140/agt.2015.15.1363⟩. ⟨hal-00872107⟩
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