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On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings



We give presentations of the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In particular, we show the volume conjecture for these knots, which states that the leading terms of the expansions present the hyperbolic volume and the Chern--Simons invariant of the complements of the knots. As higher coefficients of the expansions, we obtain a new series of invariants of these knots.

A non-trivial part of the proof is to apply the saddle point method to calculate the asymptotic expansion of an integral which presents the Kashaev invariant. A key step of this part is to give a concrete homotopy of the (real 3-dimensional) domain of the integral in ℂ3 in such a way that the boundary of the domain always stays in a certain domain in ℂ3 given by the potential function of the hyperbolic structure.



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[1] Andersen, J. E. and Hansen, S. K. Asymptotics of the quantum invariants for surgeries on the figure 8 knot. J. Knot Theory Ramifications 15 (2006), 479548.
[2] Dimofte, T. D. and Garoufalidis, S. The quantum content of the gluing equations. Geom. Topol. 17 (2013), 12531315.
[3] Dimofte, T., Gukov, S., Lenells, J. and Zagier, D. Exact results for perturbative Chern–Simons theory with complex gauge group. Commun. Number Theory Phys. 3 (2009), 363443.
[4] Dubois, J. and Kashaev, R. On the asymptotic expansion of the colored Jones polynomial for torus knots. Math. Ann. 339 (2007), 757782.
[5] Faddeev, L. D. Discrete Heisenberg–Weyl group and modular group. Lett. Math. Phys. 34 (1995), 249254.
[6] Faddeev, L. D. and Kashaev, R. M. Strongly coupled quantum discrete Liouville theory. II. Geometric interpretation of the evolution operator. J. Phys. A 35 (2002), 40434048.
[7] Faddeev, L. D., Kashaev, R. M. and Yu, A. Volkov. Strongly coupled quantum discrete Liouville theory. I. Algebraic approach and duality. Comm. Math. Phys. 219 (2001), 199219.
[8] Garoufalidis, S. and Le, T. T. Q. On the volume conjecture for small angles. arXiv:math/0502163.
[9] Gukov, S. Three-dimensional quantum gravity, Chern–Simons theory, and the A-polynomial. Comm. Math. Phys. 255 (2005), 577627.
[10] Gukov, S. and Murakami, H. SL(2, ℂ) Chern–Simons theory and the asymptotic behavior of the colored Jones polynomial. Lett. Math. Phys. 86 (2008), 7998.
[11] Hikami, K. Quantum invariant for torus link and modular forms. Comm. Math. Phys. 246 (2004), 403426.
[12] Kashaev, R. M. Quantum dilogarithm as a 6j-symbol. Modern Phys. Lett. A9 (1994), 37573768.
[13] Kashaev, R. M. A link invariant from quantum dilogarithm. Mod. Phys. Lett. A10 (1995), 14091418.
[14] Kashaev, R. M. The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39 (1997), 269275.
[15] Kashaev, R. M. and Tirkkonen, O. A proof of the volume conjecture on torus knots (Russian). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 269 (2000), Vopr. Kvant. Teor. Polya i Stat. Fiz. 16, 262268, 370; translation in J. Math. Sci. (N.Y.) 115 (2003), 2033–2036.
[16] Kashaev, R. M. and Yokota, Y. On the volume conjecture for the knot 52. preprint.
[17] Murakami, H. An introduction to the volume conjecture. Interactions between hyperbolic geometry, quantum topology and number theory, Contemp. Math. 541 (Amer. Math. Soc., Providence, RI, 2011), 140.
[18] Murakami, H. and Murakami, J. The coloured Jones polynomials and the simplicial volume of a knot. Acta Math. 186 (2001), 85104.
[19] Murakami, H., Murakami, J., Okamoto, M., Takata, T. and Yokota, Y. Kashaev's conjecture and the Chern–Simons invariants of knots and links. Experiment. Math. 11 (2002), 427435.
[20] Ohtsuki, T. On the asymptotic expansion of the Kashaev invariant of the 52 knot. Quantum Topology 7 (2016), 669735.
[21] Ohtsuki, T. On the asymptotic expansion of the Kashaev invariant of the hyperbolic knots with 7 crossings. preprint.
[22] Ohtsuki, T. and Takata, T. On the Kashaev invariant and the twisted Reidemeister torsion of two-bridge knots. Geom. Topol. 19 (2015), 853952.
[23] Stein, E. M. and Weiss, G. Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series 32 (Princeton University Press, Princeton, N.J., 1971).
[24] Takata, T. On the asymptotic expansions of the Kashaev invariant of some hyperbolic knots with 8 crossings. preprint.
[25] Thurston, D. P. Hyperbolic volume and the Jones polynomial. Notes accompanying lectures at the summer school on quantum invariants of knots and three-manifolds (Joseph Fourier Institute, University of Grenoble, org. C. Lescop, June, 1999),
[26] van der Veen, R. Proof of the volume conjecture for Whitehead chains. Acta Math. Vietnam 33 (2008), 421431.
[27] van der Veen, R. A cabling formula for the coloured Jones polynomial. arXiv:0807.2679.
[28] Yamazaki, M. and Yokota, Y. On the limit of the colored Jones polynomial of a non-simple link. Tokyo J. Math. 33 (2010), 537551.
[29] Witten, E. Analytic continuation of Chern–Simons theory. Chern–Simons gauge theory: 20 years after. AMS/IP Stud. Adv. Math., 50 (Amer. Math. Soc., Providence, RI, 2011), 347446.
[30] Wong, R. Asymptotic approximations of integrals. Computer Science and Scientific Computing (Academic Press, Inc., Boston, MA, 1989).
[31] Woronowicz, S. L. Quantum exponential function. Rev. Math. Phys. 12 (2000), 873920.
[32] Yokota, Y. On the volume conjecture for hyperbolic knots. math.QA/0009165.
[33] Yokota, Y. From the Jones polynomial to the A-polynomial of hyperbolic knots. Proceedings of the Winter Workshop of Topology/Workshop of Topology and Computer (Sendai, 2002/Nara, 2001). Interdiscip. Inform. Sci. 9 (2003), 1121.
[34] Yokota, Y. On the Kashaev invariant of twist knots. “Intelligence of Low-dimensional Topology” (edited by Ohtsuki, T. and Wakui, M.), RIMS Kokyuroku 1766 (2011), 4551.
[35] Zagier, D. Quantum modular forms. Quanta of maths. Clay Math. Proc. 11 (Amer. Math. Soc., Providence, RI, 2010), 659675.
[36] Zheng, H. Proof of the volume conjecture for Whitehead doubles of a family of torus knots. Chin. Ann. Math. Ser. B 28 (2007), 375388.


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