Quantum knot invariants

  • Stavros GaroufalidisEmail author
Part of the following topical collections:
  1. Modular Forms are Everywhere: Celebration of Don Zagier’s 65th Birthday


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.


Jones polynomial Knots Quantum topology Volume conjecture Nahm sums Stability Modularity Modular forms Mock-modular forms q-Holonomic sequence q-Series 

Mathematics Subject Classification

Primary 57N10 Secondary 57M25 



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

Ethics approval and consent to participate

Not applicable.


  1. 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. 2.
    Armond, C.: The head and tail conjecture for alternating knots. Algebr. Geom. Topol. 13(5), 2809–2826 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bar-Natan, D.: Knotatlas (2005)
  4. 4.
    Bar-Natan, D., Garoufalidis, S.: On the Melvin–Morton–Rozansky conjecture. Invent. Math. 125(1), 103–133 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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 originalGoogle Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Costantino, F.: Integrality of Kauffman brackets of trivalent graphs. Quantum Topol. 5(2), 143–184 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Culler, M.: Tables of \(A\)-polynomials (2010)
  9. 9.
    Dimofte, T., Garoufalidis, S.: Quantum Modularity and Complex Chern–Simons Theory. arXiv:1511.05628, Preprint 2015
  10. 10.
    Dunfield, N.M., Garoufalidis, S.: Incompressibility criteria for spun-normal surfaces. Trans. Am. Math. Soc. 364(11), 6109–6137 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dimofte, T., Garoufalidis, S.: The quantum content of the gluing equations. Geom. Topol. 17(3), 1253–1315 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dimofte, T., Gaiotto, D., Gukov, S.: 3-Manifolds and 3d indices. Adv. Theor. Math. Phys. 17(5), 975–1076 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dimofte, T.: Quantum Riemann surfaces in Chern–Simons theory. Adv. Theor. Math. Phys. 17(3), 479–599 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)Google Scholar
  17. 17.
    Garoufalidis, S.: Chern–Simons theory, analytic continuation and arithmetic. Acta Math. Vietnam. 33(3), 335–362 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Garoufalidis, S.: The degree of a \(q\)-holonomic sequence is a quadratic quasi-polynomial. Electron. J. Combin. 18(2), 23 (2011). Paper 4MathSciNetzbMATHGoogle Scholar
  19. 19.
    Garoufalidis, S.: The Jones slopes of a knot. Quantum Topol. 2(1), 43–69 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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)Google Scholar
  21. 21.
    Gelca, R.: On the relation between the \(A\)-polynomial and the Jones polynomial. Proc. Am. Math. Soc. 130(4), 1235–1241 (2002). electronicMathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)Google Scholar
  23. 23.
    Garoufalidis, S., Koutschan, C.: The noncommutative \(A\)-polynomial of \((-2,3, n)\) pretzel knots. Exp. Math. 21(3), 241–251 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Garoufalidis, S., Koutschan, C.: Twisting q-holonomic sequences by complex roots of unity. ISSAC, pp. 179–186 (2012)Google Scholar
  25. 25.
    Garoufalidis, S., Lê, T.T.Q.: The colored Jones function is \(q\)-holonomic. Geom. Topol. 9, 1253–1293 (2005). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Garoufalidis, S., Lê, T.T.Q.: Asymptotics of the colored Jones function of a knot. Geom. Topol. 15, 2135–2180 (2011). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Garoufalidis, S., Lê, T. T.Q.: Nahm sums, stability and the colored Jones polynomial. Res. Math. Sci. 2, Art. 1, 55 (2015)Google Scholar
  28. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 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)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Garoufalidis, S., Sun, X.: The non-commutative \(A\)-polynomial of twist knots. J. Knot Theory Ramif. 19(12), 1571–1595 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Gukov, S.: Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial. Commun. Math. Phys. 255(3), 577–627 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Garoufalidis, S., Zagier, D.: Knots and their related \(q\)-series. In preparationGoogle Scholar
  34. 34.
    Garoufalidis, S., Zagier, D.: Quantum modularity of the Kashaev invariant. In preparationGoogle Scholar
  35. 35.
    Goette, S., Zickert, C.K.: The extended Bloch group and the Cheeger–Chern–Simons class. Geom. Topol. 11, 1623–1635 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 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)Google Scholar
  37. 37.
    Haken, W.: Theorie der Normalflächen. Acta Math. 105, 245–375 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Hatcher, A.: On the boundary curves of incompressible surfaces. Pac. J. Math. 99(2), 373–377 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Hoste, J., Shanahan, P.D.: A formula for the A-polynomial of twist knots. J. Knot Theory Ramif. 13(2), 193–209 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Jantzen, J.C.: Lectures on Quantum Groups, Graduate Studies in Mathematics, vol. 6. American Mathematical Society, Providence (1996)Google Scholar
  41. 41.
    Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. (2) 126(2), 335–388 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kashaev, R.M.: The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39(3), 269–275 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Kauffman, L.H., Lins, S.L.: Temperley–Lieb Recoupling Theory and Invariants of 3-Manifolds, vol. 134. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar
  44. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Lê, T.T.Q.: The colored Jones polynomial and the \(A\)-polynomial of knots. Adv. Math. 207(2), 782–804 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 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. HuynhMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Murakami, H., Murakami, J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186(1), 85–104 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Maclachlan, C., Reid, A.W.: The Arithmetic of Hyperbolic 3-Manifolds, Graduate Texts in Mathematics, vol. 219. Springer, New York (2003)CrossRefzbMATHGoogle Scholar
  49. 49.
    Murakami, H.: Some limits of the colored Jones polynomials of the figure-eight knot. Kyungpook Math. J. 44(3), 369–383 (2004)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Neumann, W.D.: Extended Bloch group and the Cheeger–Chern–Simons class. Geom. Topol. 8, 413–474 (2004). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Nahm, W., Recknagel, A., Terhoeven, M.: Dilogarithm identities in conformal field theory. Mod. Phys. Lett. A 8(19), 1835–1847 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 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)Google Scholar
  53. 53.
    Rolfsen, D.: Knots and Links, Mathematics Lecture Series, vol. 7, Publish or Perish Inc., Houston, TX, 1990, Corrected reprint of the 1976 originalGoogle Scholar
  54. 54.
    Thurston, W.: The Geometry and Topology of 3-Manifolds, Universitext, Springer, Berlin, Lecture Notes, Princeton (1977)Google Scholar
  55. 55.
    Tran, A.T.: Proof of a stronger version of the AJ conjecture for torus knots, arXiv:1111.5065, Preprint 2012
  56. 56.
    Turaev, V.G.: The Yang–Baxter equation and invariants of links. Invent. Math. 92(3), 527–553 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 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. 58.
    Vlasenko, M., Zwegers, S.: Nahm’s conjecture: asymptotic computations and counterexamples. Commun. Number Theory Phys. 5(3), 617–642 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Witten, E.: Fivebranes and knots. Quantum Topol. 3(1), 1–137 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Zagier, D.: Vassiliev invariants and a strange identity related to the Dedekind eta-function. Topology 40(5), 945–960 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Zagier, D.: The Dilogarithm Function, Frontiers in Number Theory, Physics, and Geometry, pp. 3–65. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  64. 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. 2007/2008Google Scholar
  65. 65.
    Zagier, D.: Quantum Modular Forms, Quanta of maths, Clay Mathematics Proceedings, vol. 11, American Mathematical Society, Providence, RI, pp. 659–675 (2010)Google Scholar
  66. 66.
    Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32(3), 321–368 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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