Hostname: page-component-7857688df4-p2b9w Total loading time: 0 Render date: 2025-11-13T18:25:31.724Z Has data issue: false hasContentIssue false
  • English
  • Français

The colored Jones polynomial of the figure-eight knot and a quantum modularity

Published online by Cambridge University Press:  20 March 2023

Hitoshi Murakami Open the ORCID record for Hitoshi Murakami [Opens in a new window]
Hitoshi Murakami*
Affiliation:
Graduate School of Information Sciences, Tohoku University, Aramaki-aza-Aoba 6-3-09, Aoba-ku, Sendai 980-8579, Japan
*
e-mail hitoshi@tohoku.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the asymptotic behavior of the N-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp \bigl ((u+2p\pi \sqrt {-1})/N\bigr )$, where u is a small real number and p is a positive integer. We show that it is asymptotically equivalent to the product of the p-dimensional colored Jones polynomial evaluated at $\exp \bigl (4N\pi ^2/(u+2p\pi \sqrt {-1})\bigr )$ and a term that grows exponentially with growth rate determined by the Chern–Simons invariant. This indicates a quantum modularity of the colored Jones polynomial.

Keywords

Colored Jones polynomialvolume conjecturefigure-eight knotChern–Simons invariantReidemeister torsionquantum modularity

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 76 , Issue 2 , April 2024 , pp. 519 - 554
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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.)

Article purchase

Temporarily unavailable

Footnotes

This work was supported by JSPS KAKENHI Grant Numbers JP22H01117, JP20K03601, and JP20K03931.

References

Bettin, S. and Drappeau, S., Modularity and value distribution of quantum invariants of hyperbolic knots . Math. Ann. 382(2022), no. 34, 1631–1679.CrossRefGoogle Scholar
Dimofte, T. and Gukov, S., Quantum field theory and the volume conjecture . In: Interactions between hyperbolic geometry, quantum topology and number theory, Contemporary Mathematics, 541, American Mathematical Society, Providence, RI, 2011, pp. 41–67.CrossRefGoogle Scholar
Faddeev, L. D., Discrete Heisenberg–Weyl group and modular group . Lett. Math. Phys. 34(1995), no. 3, 249–254.CrossRefGoogle Scholar
Garoufalidis, S. and Le, T. T. Q., An analytic version of the Melvin–Morton–Rozansky conjecture. Preprint, 2005. arXiv:math.GT/0503641 Google Scholar
Garoufalidis, S. and Le, T. T. Q., On the volume conjecture for small angles. Preprint, 2005. arXiv:math.GT/0502163 Google Scholar
Garoufalidis, S. and Lê, T. T. Q., Asymptotics of the colored Jones function of a knot . Geom. Topol. 15(2011), no. 4, 2135–2180.CrossRefGoogle Scholar
Garoufalidis, S. and Zagier, D., Knots, perturbative series and quantum modularity. Preprint, 2021. arXiv:2111.06645 Google Scholar
Gromov, M., Volume and bounded cohomology . Publ. Math. Inst. Hautes Etudes Sci. 56(1982), 5–99.Google Scholar
Gukov, S. and Murakami, H., $\mathit{\mathsf{SL}}\left(2,\mathbb{C}\right)$ Chern–Simons theory and the asymptotic behavior of the colored Jones polynomial . In: Modular forms and string duality, Fields Institute Communications, 54, American Mathematical Society, Providence, RI, 2008, pp. 261–277.Google Scholar
Habiro, K., On the colored Jones polynomials of some simple links . Sūrikaisekikenkyūsho Kōkyūroku 1172(2000), 34–43, Recent progress towards the volume conjecture (Japanese) (Kyoto, 2000).Google Scholar
Jones, V. F. R., A polynomial invariant for knots via von Neumann algebras . Bull. Amer. Math. Soc. (N.S.) 12(1985), no. 1, 103–111.CrossRefGoogle Scholar
Kashaev, R. M., A link invariant from quantum dilogarithm . Modern Phys. Lett. A 10(1995), no. 19, 1409–1418.CrossRefGoogle Scholar
Kashaev, R. M., The hyperbolic volume of knots from the quantum dilogarithm . Lett. Math. Phys. 39(1997), no. 3, 269–275.CrossRefGoogle Scholar
Kashaev, R. M. and Tirkkonen, O., A proof of the volume conjecture on torus knots . Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 269(2000), no. Vopr. Kvant. Teor. Polya i Stat. Fiz. 16, 262–268, 370.Google Scholar
Kirby, R. and Melvin, P., The $\textit{3}$ -manifold invariants of Witten and Reshetikhin–Turaev for $\mathsf{sl}(2,\mathsf{C})$ . Invent. Math. 105(1991), no. 3, 473–545.CrossRefGoogle Scholar
Kirillov, A. N. and Reshetikhin, N. Y., Representations of the algebra Uq (sl(2)), q-orthogonal polynomials and invariants of links . In: Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), Advances in Mathematical Physics, 7, World Scientific Publishing, Teaneck, NJ, 1989, pp. 285–339.Google Scholar
Le, T. T. Q., Quantum invariants of 3-manifolds: Integrality, splitting, and perturbative expansion . Topology Appl. 127(2003), nos. 1–2, 125–152.CrossRefGoogle Scholar
Le, T. T. Q. and Tran, A. T., On the volume conjecture for cables of knots . J. Knot Theory Ramifications 19(2010), no. 12, 1673–1691.CrossRefGoogle Scholar
Masbaum, G., Skein-theoretical derivation of some formulas of Habiro . Algebra Geom. Topol. 3(2003), 537–556.CrossRefGoogle Scholar
Maximon, L. C., The dilogarithm function for complex argument. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 459(2003), no. 2039, 2807–2819.CrossRefGoogle Scholar
Meyerhoff, R., Density of the Chern–Simons invariant for hyperbolic $\textit{3}$ -manifolds . In: Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), London Mathematical Society Lecture Note Series, 112, Cambridge University Press, Cambridge, 1986, pp. 217–239.Google Scholar
Milnor, J., Hyperbolic geometry: the first 150 years . Bull. Amer. Math. Soc. (N.S.) 6(1982), no. 1, 9–24.CrossRefGoogle Scholar
Murakami, H., The colored Jones polynomials and the Alexander polynomial of the figure-eight knot. J. Geom. Topol. 7(2007), no. 2, 249–269.Google Scholar
Murakami, H., The coloured Jones polynomial, the Chern–Simons invariant, and the Reidemeister torsion of the figure-eight knot . J. Topol. 6(2013), no. 1, 193–216.CrossRefGoogle Scholar
Murakami, H. and Murakami, J., The colored Jones polynomials and the simplicial volume of a knot . Acta Math. 186(2001), no. 1, 85–104.CrossRefGoogle Scholar
Murakami, H., Murakami, J., Okamoto, M., Takata, T., and Yokota, Y., Kashaev’s conjecture and the Chern–Simons invariants of knots and links . Exp. Math. 11(2002), no. 3, 427–435.CrossRefGoogle Scholar
Murakami, H. and Tran, A. T., On the asymptotic behavior of the colored Jones polynomial of the figure-eight knot associated with a real number. Preprint, 2021. arXiv:2109.04664 [math.GT]Google Scholar
Murakami, H. and Yokota, Y., The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces . J. Reine Angew. Math. 607(2007), 47–68.Google Scholar
Murakami, H. and Yokota, Y., Volume conjecture for knots, SpringerBriefs in Mathematical Physics, 30, Springer, Singapore, 2018.CrossRefGoogle Scholar
Neumann, W. D. and Zagier, D., Volumes of hyperbolic three-manifolds . Topology 24(1985), no. 3, 307–332.CrossRefGoogle Scholar
Ohtsuki, T., On the asymptotic expansion of the Kashaev invariant of the ${5}_2$ knot . Quantum Topol. 7(2016), no. 4, 669–735.CrossRefGoogle Scholar
Ohtsuki, T., On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings . Int. J. Math. 28(2017), no. 13, 1750096, 143.CrossRefGoogle Scholar
Ohtsuki, T. and Yokota, Y., On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings . Math. Proc. Cambridge Philos. Soc. 165(2018), no. 2, 287–339.CrossRefGoogle Scholar
Reshetikhin, N. Y. and Turaev, V. G., Ribbon graphs and their invariants derived from quantum groups . Comm. Math. Phys. 127(1990), no. 1, 1–26.CrossRefGoogle Scholar
Thurston, W. P., The geometry and topology of three-manifolds/William P. Thurston; with a preface by Steven P. Kerckhoff, American Mathematical Society, Providence, RI, 2022.Google Scholar
Zagier, D., Quantum modular forms . In: Quanta of maths, Clay Mathematics Proceedings, 11, American Mathematical Society, Providence, RI, 2010, pp. 659–675.Google Scholar
Zheng, H., Proof of the volume conjecture for whitehead doubles of a family of torus knots . Chinese Ann. Math. Ser. B 28(2007), no. 4, 375–388.CrossRefGoogle Scholar