Relations between Witten–Reshetikhin–Turaev and nonsemisimple sl(2) 3–manifold invariants

Costantino, Francesco¹ ; Geer, Nathan² ; Patureau-Mirand, Bertrand³

¹Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France
²Mathematics and Statistics, Utah State University, 3900 Old Main Hill, Logan, UT 84322-3900, USA
³UMR 6205, LMBA, Université de Bretagne-Sud, Campus de Tohannic, BP 573, 56017 Vannes, France

Algebraic and Geometric Topology, Tome 15 (2015) no. 3, pp. 1363-1386

Résumé

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.

Détail

DOI : 10.2140/agt.2015.15.1363

Classification : 57N10, 57R56
Keywords: quantum invariants, Reshetikhin-Turaev invariants, Hennings invariants, $3$–manifolds

Bibliographie

[1] Y Akutsu, T Deguchi, T Ohtsuki, Invariants of colored links, J. Knot Theory Ramifications 1 (1992) 161

[2] C Blanchet, Invariants on three-manifolds with spin structure, Comment. Math. Helv. 67 (1992) 406

[3] C Blanchet, F Costantino, N Geer, B Patureau-Mirand, Non semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants,

[4] C Blanchet, N Habegger, G Masbaum, P Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology 31 (1992) 685

[5] Q Chen, S Kuppum, P Srinivasan, On the relation between the WRT invariant and the Hennings invariant, Math. Proc. Cambridge Philos. Soc. 146 (2009) 151

[6] F Costantino, N Geer, B Patureau-Mirand, Quantum invariants of $3$–manifolds via link surgery presentations and non-semi-simple categories, J. Topol. 7 (2014) 1005

[7] F Costantino, N Geer, B Patureau-Mirand, Some remarks on the unrolled quantum group of $\mathfrak{sl}(2)$, J. Pure Appl. Algebra 219 (2015) 3238

[8] F Costantino, J Murakami, On the $\mathrm{SL}(2,\mathbb{C})$ quantum $6j$–symbols and their relation to the hyperbolic volume, Quantum Topol. 4 (2013) 303

[9] N Geer, B Patureau-Mirand, V Turaev, Modified quantum dimensions and re-normalized link invariants, Compos. Math. 145 (2009) 196

[10] M Hennings, Invariants of links and $3$–manifolds obtained from Hopf algebras, J. London Math. Soc. 54 (1996) 594

[11] L H Kauffman, D E Radford, Invariants of $3$–manifolds derived from finite-dimensional Hopf algebras, J. Knot Theory Ramifications 4 (1995) 131

[12] T Kerler, Homology TQFT's and the Alexander–Reidemeister invariant of $3$–manifolds via Hopf algebras and skein theory, Canad. J. Math. 55 (2003) 766

[13] R Kirby, P Melvin, The $3$–manifold invariants of Witten and Reshetikhin–Turaev for $\mathrm{sl}(2,\mathbf{C})$, Invent. Math. 105 (1991) 473

[14] J Murakami, Generalized Kashaev invariants for knots in three manifolds,

[15] T Ohtsuki, Quantum invariants: a study of knots, $3$–manifolds, and their sets, Series on Knots and Everything 29, World Scientific (2002)

[16] N Reshetikhin, V G Turaev, Invariants of $3$–manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547

[17] V G Turaev, Quantum invariants of knots and $3$–manifolds, Studies in Mathematics 18, de Gruyter (1994)

[18] E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351