Skip to main content
SpringerLink
Log in

Quantum knot invariants

  • Research
  • Published:
Research in the Mathematical Sciences Aims and scope Submit manuscript

Abstract

This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective computations. The talk was given in the Mathematische Arbeitstagung June 24–July 1, 2011.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Armond, C., Dasbach, O.: Rogers–Ramanujan type identities and the head and tail of the colored jones polynomial (2011) arXiv:1106.3948, Preprint

  2. Armond, C.: The head and tail conjecture for alternating knots. Algebr. Geom. Topol. 13(5), 2809–2826 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bar-Natan, D.: Knotatlas (2005) http://katlas.org

  4. Bar-Natan, D., Garoufalidis, S.: On the Melvin–Morton–Rozansky conjecture. Invent. Math. 125(1), 103–133 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bender, C.M., Orszag, S.A.: Advanced mathematical methods for scientists and engineers. I. Springer, New York (1999) Asymptotic methods and perturbation theory, Reprint of the 1978 original

  6. Cooper, D., Culler, M., Gillet, H., Long, D.D., Shalen, P.B.: Plane curves associated to character varieties of 3-manifolds. Invent. Math. 118(1), 47–84 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Costantino, F.: Integrality of Kauffman brackets of trivalent graphs. Quantum Topol. 5(2), 143–184 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Culler, M.: Tables of \(A\)-polynomials (2010) http://www.math.uic.edu/~culler/Apolynomials

  9. Dimofte, T., Garoufalidis, S.: Quantum Modularity and Complex Chern–Simons Theory. arXiv:1511.05628, Preprint 2015

  10. Dunfield, N.M., Garoufalidis, S.: Incompressibility criteria for spun-normal surfaces. Trans. Am. Math. Soc. 364(11), 6109–6137 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dimofte, T., Garoufalidis, S.: The quantum content of the gluing equations. Geom. Topol. 17(3), 1253–1315 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dimofte, T., Gaiotto, D., Gukov, S.: 3-Manifolds and 3d indices. Adv. Theor. Math. Phys. 17(5), 975–1076 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dimofte, T., Gukov, S., Lenells, J., Zagier, D.: Exact results for perturbative Chern–Simons theory with complex gauge group. Commun. Number Theory Phys. 3(2), 363–443 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dimofte, T.: Quantum Riemann surfaces in Chern–Simons theory. Adv. Theor. Math. Phys. 17(3), 479–599 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Futer, D., Kalfagianni, E., Purcell, J.S.: Slopes and colored Jones polynomials of adequate knots. Proc. Am. Math. Soc. 139(5), 1889–1896 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Garoufalidis, S.: On the characteristic and deformation varieties of a knot. In: Proceedings of the Casson Fest, Geometry and Topology Monographs, vol. 7, Geometry and Topology Public, Coventry (2004), pp. 291–309 (electronic)

  17. Garoufalidis, S.: Chern–Simons theory, analytic continuation and arithmetic. Acta Math. Vietnam. 33(3), 335–362 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Garoufalidis, S.: The degree of a \(q\)-holonomic sequence is a quadratic quasi-polynomial. Electron. J. Combin. 18(2), 23 (2011). Paper 4

    MathSciNet  MATH  Google Scholar 

  19. Garoufalidis, S.: The Jones slopes of a knot. Quantum Topol. 2(1), 43–69 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Garoufalidis, S.: Knots and tropical curves. In: Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory. Contemporary Mathematics, vol. 541, American Mathematical Society, Providence, RI, pp. 83–101 (2011)

  21. Gelca, R.: On the relation between the \(A\)-polynomial and the Jones polynomial. Proc. Am. Math. Soc. 130(4), 1235–1241 (2002). electronic

    Article  MathSciNet  MATH  Google Scholar 

  22. Garoufalidis, S., Its, A., Kapaev, A., Mariño, M.: Asymptotics of the instantons of Painlevé I. Int. Math. Res. Not. IMRN, no. 3, 561–606 (2012)

  23. Garoufalidis, S., Koutschan, C.: The noncommutative \(A\)-polynomial of \((-2,3, n)\) pretzel knots. Exp. Math. 21(3), 241–251 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Garoufalidis, S., Koutschan, C.: Twisting q-holonomic sequences by complex roots of unity. ISSAC, pp. 179–186 (2012)

  25. Garoufalidis, S., Lê, T.T.Q.: The colored Jones function is \(q\)-holonomic. Geom. Topol. 9, 1253–1293 (2005). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  26. Garoufalidis, S., Lê, T.T.Q.: Asymptotics of the colored Jones function of a knot. Geom. Topol. 15, 2135–2180 (2011). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  27. Garoufalidis, S., Lê, T. T.Q.: Nahm sums, stability and the colored Jones polynomial. Res. Math. Sci. 2, Art. 1, 55 (2015)

  28. Gukov, S., Murakami, H.: \(\text{ SL }(2, \mathbb{C})\) Chern–Simons theory and the asymptotic behavior of the colored Jones polynomial. Lett. Math. Phys. 86(2–3), 79–98 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Garoufalidis, S., Mattman, T.W.: The \(A\)-polynomial of the \((-2,3,3+2n)\) pretzel knots. New York J. Math. 17, 269–279 (2011)

    MathSciNet  MATH  Google Scholar 

  30. Garoufalidis, S., Sun, X.: The non-commutative \(A\)-polynomial of twist knots. J. Knot Theory Ramif. 19(12), 1571–1595 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gukov, S.: Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial. Commun. Math. Phys. 255(3), 577–627 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Garoufalidis, S., van der Veen, R.: Asymptotics of quantum spin networks at a fixed root of unity. Math. Ann. 352(4), 987–1012 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Garoufalidis, S., Zagier, D.: Knots and their related \(q\)-series. In preparation

  34. Garoufalidis, S., Zagier, D.: Quantum modularity of the Kashaev invariant. In preparation

  35. Goette, S., Zickert, C.K.: The extended Bloch group and the Cheeger–Chern–Simons class. Geom. Topol. 11, 1623–1635 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Habiro, K.: On the quantum \({{\rm sl}}_2\) invariants of knots and integral homology spheres, Invariants of knots and 3-manifolds (Kyoto, 2001), Geometry and Topology Monographs, vol. 4, Geometry and Topology Publications, Coventry, 2002, pp. 55–68 (electronic)

  37. Haken, W.: Theorie der Normalflächen. Acta Math. 105, 245–375 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hatcher, A.: On the boundary curves of incompressible surfaces. Pac. J. Math. 99(2), 373–377 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hoste, J., Shanahan, P.D.: A formula for the A-polynomial of twist knots. J. Knot Theory Ramif. 13(2), 193–209 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Jantzen, J.C.: Lectures on Quantum Groups, Graduate Studies in Mathematics, vol. 6. American Mathematical Society, Providence (1996)

  41. Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. (2) 126(2), 335–388 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kashaev, R.M.: The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39(3), 269–275 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kauffman, L.H., Lins, S.L.: Temperley–Lieb Recoupling Theory and Invariants of 3-Manifolds, vol. 134. Princeton University Press, Princeton (1994)

    MATH  Google Scholar 

  44. Kontsevich, M., Soibelman, Y.: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants. Commun. Number Theory Phys. 5(2), 231–352 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lê, T.T.Q.: The colored Jones polynomial and the \(A\)-polynomial of knots. Adv. Math. 207(2), 782–804 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  46. Le, T.T.Q., Tran, A.T.: On the AJ conjecture for knots. Indiana Univ. Math. J. 64(4), 1103–1151 (2015). With an appendix written jointly with Vu Q. Huynh

    Article  MathSciNet  MATH  Google Scholar 

  47. Murakami, H., Murakami, J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186(1), 85–104 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. Maclachlan, C., Reid, A.W.: The Arithmetic of Hyperbolic 3-Manifolds, Graduate Texts in Mathematics, vol. 219. Springer, New York (2003)

    Book  MATH  Google Scholar 

  49. Murakami, H.: Some limits of the colored Jones polynomials of the figure-eight knot. Kyungpook Math. J. 44(3), 369–383 (2004)

    MathSciNet  MATH  Google Scholar 

  50. Neumann, W.D.: Extended Bloch group and the Cheeger–Chern–Simons class. Geom. Topol. 8, 413–474 (2004). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  51. Nahm, W., Recknagel, A., Terhoeven, M.: Dilogarithm identities in conformal field theory. Mod. Phys. Lett. A 8(19), 1835–1847 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  52. Petkovšek, M., Wilf, H.S., Zeilberger, D.: \(A=B\), A K Peters Ltd., Wellesley, MA, 1996, With a foreword by Donald E. Knuth, With a separately available computer disk (1996)

  53. Rolfsen, D.: Knots and Links, Mathematics Lecture Series, vol. 7, Publish or Perish Inc., Houston, TX, 1990, Corrected reprint of the 1976 original

  54. Thurston, W.: The Geometry and Topology of 3-Manifolds, Universitext, Springer, Berlin, Lecture Notes, Princeton (1977)

  55. Tran, A.T.: Proof of a stronger version of the AJ conjecture for torus knots, arXiv:1111.5065, Preprint 2012

  56. Turaev, V.G.: The Yang–Baxter equation and invariants of links. Invent. Math. 92(3), 527–553 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  57. Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studies in Mathematics, vol. 18. Walter de Gruyter & Co, Berlin (1994)

    Google Scholar 

  58. Vlasenko, M., Zwegers, S.: Nahm’s conjecture: asymptotic computations and counterexamples. Commun. Number Theory Phys. 5(3), 617–642 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  59. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  60. Witten, E.: Fivebranes and knots. Quantum Topol. 3(1), 1–137 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  61. Wilf, H.S., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and “\(q\)”) multisum/integral identities. Invent. Math. 108(3), 575–633 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  62. Zagier, D.: Vassiliev invariants and a strange identity related to the Dedekind eta-function. Topology 40(5), 945–960 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  63. Zagier, D.: The Dilogarithm Function, Frontiers in Number Theory, Physics, and Geometry, pp. 3–65. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  64. Zagier, D.: Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann), Astérisque (2009), no. 326, Exp. No. 986, vii–viii, 143–164 (2010), Séminaire Bourbaki. Vol. 2007/2008

  65. Zagier, D.: Quantum Modular Forms, Quanta of maths, Clay Mathematics Proceedings, vol. 11, American Mathematical Society, Providence, RI, pp. 659–675 (2010)

  66. Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32(3), 321–368 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author was supported in part by NSF. To Don Zagier, with admiration.

Ethics approval and consent to participate

Not applicable.

Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Stavros Garoufalidis

Authors
  1. Stavros Garoufalidis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stavros Garoufalidis.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Garoufalidis, S. Quantum knot invariants. Res Math Sci 5, 11 (2018). https://doi.org/10.1007/s40687-018-0127-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40687-018-0127-3

Keywords

Mathematics Subject Classification

Search

Navigation