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Relations between Witten–Reshetikhin–Turaev and nonsemisimple $\mathfrak{sl}(2)$ $3$–manifold invariants

Francesco Costantino, Nathan Geer and Bertrand Patureau-Mirand

Algebraic & Geometric Topology 15 (2015) 1363–1386

The Witten–Reshetikhin–Turaev (WRT) invariants extend the Jones polynomials of links in S3 to invariants of links in 3–manifolds. Similarly, the authors constructed two 3–manifold invariants Nr and Nr0 which extend the Akutsu–Deguchi–Ohtsuki (ADO) invariant of links in S3 colored by complex numbers to links in arbitrary manifolds. All these invariants are based on the representation theory of the quantum group Uqsl2, where the definition of the invariants Nr and Nr0 uses a nonstandard category of Uqsl2–modules which is not semisimple. In this paper we study the second invariant, Nr0, and consider its relationship with the WRT invariants. In particular, we show that the ADO invariant of a knot in S3 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 Nr0 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 S3 or a connected sum of such manifolds.

quantum invariants, Reshetikhin-Turaev invariants, Hennings invariants, $3$–manifolds
Mathematical Subject Classification 2010
Primary: 57N10
Secondary: 57R56
Received: 10 October 2013
Revised: 26 June 2014
Accepted: 5 October 2014
Published: 19 June 2015
Francesco Costantino
Institut de Mathématiques de Toulouse
118 route de Narbonne
31062 Toulouse
Nathan Geer
Mathematics and Statistics
Utah State University
3900 Old Main Hill
Logan, UT 84322-3900
Bertrand Patureau-Mirand
UMR 6205, LMBA
Université de Bretagne-Sud
Campus de Tohannic, BP 573
56017 Vannes