Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-03T01:35:03.146Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 June 2017

TOMOTADA OHTSUKI  and
YOSHIYUKI YOKOTA
TOMOTADA OHTSUKI
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan. e-mail: [email protected]
YOSHIYUKI YOKOTA
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Tokyo, 192-0397, Japan. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 2 , September 2018 , pp. 287 - 339
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[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), 479–548.Google Scholar
[2] Dimofte, T. D. and Garoufalidis, S. The quantum content of the gluing equations. Geom. Topol. 17 (2013), 1253–1315.Google Scholar
[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), 363–443.Google Scholar
[4] Dubois, J. and Kashaev, R. On the asymptotic expansion of the colored Jones polynomial for torus knots. Math. Ann. 339 (2007), 757–782.Google Scholar
[5] Faddeev, L. D. Discrete Heisenberg–Weyl group and modular group. Lett. Math. Phys. 34 (1995), 249–254.Google Scholar
[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), 4043–4048.Google Scholar
[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), 199–219.Google Scholar
[8] Garoufalidis, S. and Le, T. T. Q. On the volume conjecture for small angles. arXiv:math/0502163.Google Scholar
[9] Gukov, S. Three-dimensional quantum gravity, Chern–Simons theory, and the A-polynomial. Comm. Math. Phys. 255 (2005), 577–627.Google Scholar
[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), 79–98.Google Scholar
[11] Hikami, K. Quantum invariant for torus link and modular forms. Comm. Math. Phys. 246 (2004), 403–426.Google Scholar
[12] Kashaev, R. M. Quantum dilogarithm as a 6j-symbol. Modern Phys. Lett. A9 (1994), 3757–3768.Google Scholar
[13] Kashaev, R. M. A link invariant from quantum dilogarithm. Mod. Phys. Lett. A10 (1995), 1409–1418.Google Scholar
[14] Kashaev, R. M. The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39 (1997), 269–275.Google Scholar
[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, 262–268, 370; translation in J. Math. Sci. (N.Y.) 115 (2003), 2033–2036.Google Scholar
[16] Kashaev, R. M. and Yokota, Y. On the volume conjecture for the knot 52. preprint.Google Scholar
[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), 1–40.Google Scholar
[18] Murakami, H. and Murakami, J. The coloured Jones polynomials and the simplicial volume of a knot. Acta Math. 186 (2001), 85–104.Google Scholar
[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), 427–435.Google Scholar
[20] Ohtsuki, T. On the asymptotic expansion of the Kashaev invariant of the 52 knot. Quantum Topology 7 (2016), 669–735.Google Scholar
[21] Ohtsuki, T. On the asymptotic expansion of the Kashaev invariant of the hyperbolic knots with 7 crossings. preprint.Google Scholar
[22] Ohtsuki, T. and Takata, T. On the Kashaev invariant and the twisted Reidemeister torsion of two-bridge knots. Geom. Topol. 19 (2015), 853–952.Google Scholar
[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).Google Scholar
[24] Takata, T. On the asymptotic expansions of the Kashaev invariant of some hyperbolic knots with 8 crossings. preprint.Google Scholar
[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), http://www.math.columbia.edu/~dpt/speaking/Grenoble.pdf.Google Scholar
[26] van der Veen, R. Proof of the volume conjecture for Whitehead chains. Acta Math. Vietnam 33 (2008), 421–431.Google Scholar
[27] van der Veen, R. A cabling formula for the coloured Jones polynomial. arXiv:0807.2679.Google Scholar
[28] Yamazaki, M. and Yokota, Y. On the limit of the colored Jones polynomial of a non-simple link. Tokyo J. Math. 33 (2010), 537–551.Google Scholar
[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), 347–446.Google Scholar
[30] Wong, R. Asymptotic approximations of integrals. Computer Science and Scientific Computing (Academic Press, Inc., Boston, MA, 1989).Google Scholar
[31] Woronowicz, S. L. Quantum exponential function. Rev. Math. Phys. 12 (2000), 873–920.Google Scholar
[32] Yokota, Y. On the volume conjecture for hyperbolic knots. math.QA/0009165.Google Scholar
[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), 11–21.Google Scholar
[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), 45–51.Google Scholar
[35] Zagier, D. Quantum modular forms. Quanta of maths. Clay Math. Proc. 11 (Amer. Math. Soc., Providence, RI, 2010), 659–675.Google Scholar
[36] Zheng, H. Proof of the volume conjecture for Whitehead doubles of a family of torus knots. Chin. Ann. Math. Ser. B 28 (2007), 375–388.Google Scholar