Abstract
In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern–Simons invariant and twisted Reidemeister torsion with coefficients in the adjoint representation.
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References
Burde G. (1967). Darstellungen von Knotengruppen (German). Math. Ann. 173: 24–33
Culler M. and Shalen P. (1983). Varieties of group representations and splittings of 3-manifolds. Ann. Math. 117: 109–146
Rham G. (1967). Introduction aux polynmes d’un nœ ud (French). Enseign. Math. 13(2): 187–194
Dubois J. (2005). Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups. Ann. Inst Fourier 55: 1685–1734
Dubois, J.: Non abelian twisted Reidemeister torsion for fibered knots. Can. Math. Bull. 49, 55–71 (2006)
Heusener M., Porti J. and Suárez E. (2001). Deformations of reducible representations of 3-manifold groups into SL2(\({\mathbb{C}}\)). J. reine angew. Math. 530: 191–227
Hikami K. (2003). Volume Conjecture and asymptotic expansion of q-series. Exp. Math. 12: 319–337
Hikami K. and Kirillov A.N. (2003). Torus knot and minimal model. Phys. Lett. B 575: 343–348
Kashaev R. (1997). The hyperbolic volume of knots from quantum dilogarithm. Lett. Math. Phys. 39: 269–275
Kashaev R. and Tirkkonen O. (2003). Proof of the volume conjecture for torus knots. J. Math. Sci. N. Y. 115: 2033–2036
Kirk P. and Klassen E. (1993). Chern–Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space of T 2. Commun. Math. Phys. 153: 521–557
Klassen E. (1991). Representations of knot groups in SU(2). Trans. Am. Math. Soc. 326: 795–828
Le T. (1994). Varieties of representations and their subvarieties of cohomology jumps for certain knot groups. Russ. Acad. Sci. Sb. Math. 78: 187–209
Milnor J. (1962). A duality theorem for Reidemeister Torsion. Ann. Math. 76: 134–147
Milnor J. (1966). Whitehead torsion. Bull. Am. Math. Soc. 72: 358–426
Murakami H. and Murakami J. (2001). The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186: 85–104
Murakami H. (2004). Asymptotic behaviors of the colored Jones polynomials of a torus knot. Int. J. Math. 15: 547–555
Ohtsuki T. (2002). Problems on Invariants of Knots and 3-Manifolds. Geom. Topol. Monogr. 4: 377–572
Porti, J.: Torsion de Reidemeister pour les variétés hyperboliques. Mem. Am. Math. Soc. 128(612), x + 139 pp (1997)
Riley R. (1984). Nonabelian representations of 2-bridge knot groups. Quart. J. Math. Oxf. 35: 191–208
Ramadas T., Singer I. and Weitsman J. (1989). Some comments on Chern–Simons gauge theory. Comm. Math. Phys. 126: 409–420
Turaev V. (1986). Reidemeister torsion in knot theory, English version. Russ. Math. Surv. 41: 119–182
Turaev V. (2001). Introduction to combinatorial torsions. Birkhäuser, Basel
Turaev V. (2002). Torsions of 3-dimensional manifolds. Birkhäuser, Basel
Yamaguchi, Y.: The limit values of the non-abelian twisted Reidemeister torsion associated to knots. Preprint arXiv:math.GT/0512277 (2005)
Zheng, H.: Proof of the volume conjecture for Whitehead double of tours knots. Preprint available at arXiv:math.GT/0508138
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This work is supported in part by the Swiss National Science Foundation, the first author (J. Dubois) is also supported by the European Community with Marie Curie Intra–European Fellowship (MEIF–CT–2006–025316). While writing the paper, J. Dubois visited the CRM. He thanks the CRM for its hospitality.
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Dubois, J., Kashaev, R. On the asymptotic expansion of the colored Jones polynomial for torus knots. Math. Ann. 339, 757–782 (2007). https://doi.org/10.1007/s00208-007-0109-z
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DOI: https://doi.org/10.1007/s00208-007-0109-z