Abstract
The Witten–Reshetikhin–Turaev (WRT) invariants extend the Jones polynomials of links in to invariants of links in –manifolds. Similarly, the authors constructed two –manifold invariants and which extend the Akutsu–Deguchi–Ohtsuki (ADO) invariant of links in colored by complex numbers to links in arbitrary manifolds. All these invariants are based on the representation theory of the quantum group , where the definition of the invariants and uses a nonstandard category of –modules which is not semisimple. In this paper we study the second invariant, , and consider its relationship with the WRT invariants. In particular, we show that the ADO invariant of a knot in 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 and a WRT invariant. We prove this conjecture when the –manifold is not a rational homology sphere, and when is a rational homology sphere obtained by surgery on a knot in or a connected sum of such manifolds.
Citation
Francesco Costantino. Nathan Geer. Bertrand Patureau-Mirand. "Relations between Witten–Reshetikhin–Turaev and nonsemisimple $\mathfrak{sl}(2)$ $3$–manifold invariants." Algebr. Geom. Topol. 15 (3) 1363 - 1386, 2015. https://doi.org/10.2140/agt.2015.15.1363
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