Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 082, 39 pages      arXiv:2304.09377      https://doi.org/10.3842/SIGMA.2023.082

Knots and Their Related $q$-Series

Stavros Garoufalidis a and Don Zagier bc
a) International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology, Shenzhen, P.R. China
b) Max Planck Institute for Mathematics, Bonn, Germany
c) International Centre for Theoretical Physics, Trieste, Italy

Received April 25, 2023, in final form October 17, 2023; Published online November 01, 2023

Abstract
We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties that defines a ${\rm PSL}_2({\mathbb Z})$-cocycle on the space of matrix-valued piecewise analytic functions on the real numbers. We identify the corresponding cocycle with the one coming from the Kashaev invariant of a knot (and its matrix-valued extension) via the refined quantum modularity conjecture of [arXiv:2111.06645] and also relate the matrix-valued invariant with the 3D-index of Dimofte-Gaiotto-Gukov. The cocycle also has an analytic extendability property that leads to the notion of a matrix-valued holomorphic quantum modular form. This is a tale of several independent discoveries, both empirical and theoretical, all illustrated by the three simplest hyperbolic knots.

Key words: $q$-series; Nahm sums; knots; Jones polynomial; Kashaev invariant; volume conjecture; hyperbolic 3-manifolds; quantum topology; quantum modular forms; holomorphic quantum modular forms; state integrals; 3D-index; quantum dilogarithm; asymptotics; Chern-Simons theory.

pdf (760 kb)   tex (82 kb)  

References

  1. An N., Li Y., On the quantum modularity conjecture for the $(-2,3,7)$-pretzel knot, in preparation.
  2. Andersen J.E., Garoufalidis S., Kashaev R., The volume conjecture for the KLV state-integral, in preparation.
  3. Andersen J.E., Hansen S.K., Asymptotics of the quantum invariants for surgeries on the figure 8 knot, J. Knot Theory Ramifications 15 (2006), 479-548, arXiv:math.QA/0506456.
  4. Andersen J.E., Kashaev R., A TQFT from quantum Teichmüller theory, Comm. Math. Phys. 330 (2014), 887-934, arXiv:1109.6295.
  5. Beem C., Dimofte T., Pasquetti S., Holomorphic blocks in three dimensions, J. High Energy Phys. 2014 (2014), no. 12, 177, 119 pages, arXiv:1211.1986.
  6. Bettin S., Drappeau S., Modularity and value distribution of quantum invariants of hyperbolic knots, Math. Ann. 382 (2022), 1631-1679, arXiv:1905.02045.
  7. Bruinier J.H., van der Geer G., Harder G., Zagier D., The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008.
  8. Calegari F., Garoufalidis S., Zagier D., Bloch groups, algebraic $K$-theory, units, and Nahm's conjecture, Ann. Sci. Éc. Norm. Supér. 56 (2023), 383-426, arXiv:1712.04887.
  9. Dimofte T., Complex Chern-Simons theory at level $k$ via the 3d-3d correspondence, Comm. Math. Phys. 339 (2015), 619-662, arXiv:1409.0857.
  10. Dimofte T., Perturbative and nonperturbative aspects of complex Chern-Simons theory, J. Phys. A 50 (2017), 443009, 25 pages, arXiv:1608.02961.
  11. Dimofte T., Gaiotto D., Gukov S., 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013), 975-1076, arXiv:1112.5179.
  12. Dimofte T., Gaiotto D., Gukov S., Gauge theories labelled by three-manifolds, Comm. Math. Phys. 325 (2014), 367-419, arXiv:1108.4389.
  13. Dimofte T., Garoufalidis S., The quantum content of the gluing equations, Geom. Topol. 17 (2013), 1253-1315, arXiv:1202.6268.
  14. Dimofte T., Garoufalidis S., Quantum modularity and complex Chern-Simons theory, Commun. Number Theory Phys. 12 (2018), 1-52, arXiv:1511.05628.
  15. Dimofte T., Gukov S., Lenells J., Zagier D., Exact results for perturbative Chern-Simons theory with complex gauge group, Commun. Number Theory Phys. 3 (2009), 363-443, arXiv:0903.2472.
  16. Ekholm T., Gruen A., Gukov S., Kucharski P., Park S., Sułkowski P., $\widehat{Z}$ at large $N$: from curve counts to quantum modularity, Comm. Math. Phys. 396 (2022), 143-186, arXiv:2005.13349.
  17. Faddeev L., Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995), 249-254, arXiv:hep-th/9504111.
  18. Garoufalidis S., The degree of a $q$-holonomic sequence is a quadratic quasi-polynomial, Electron. J. Combin. 18 (2011), 4, 23 pages, arXiv:1005.4580.
  19. Garoufalidis S., The Jones slopes of a knot, Quantum Topol. 2 (2011), 43-69, arXiv:0911.3627.
  20. Garoufalidis S., The 3D index of an ideal triangulation and angle structures (with an appendix by Sander Zwegers), Ramanujan J. 40 (2016), 573-604, arXiv:1208.1663.
  21. Garoufalidis S., Quantum knot invariants, Res. Math. Sci. 5 (2018), 11, 17 pages, arXiv:1201.3314.
  22. Garoufalidis S., Gu J., Mariño M., The resurgent structure of quantum knot invariants, Comm. Math. Phys. 386 (2021), 469-493, arXiv:2007.10190.
  23. Garoufalidis S., Gu J., Mariño M., Peacock patterns and resurgence in complex Chern-Simons theory, Res. Math. Sci. 10 (2023), 29, 67 pages, arXiv:2012.00062.
  24. Garoufalidis S., Gu J., Mariño M., Wheeler C., Resurgence of Chern-Simons theory at the trivial flat connection, arXiv:2111.04763.
  25. Garoufalidis S., Hodgson C.D., Rubinstein J.H., Segerman H., 1-efficient triangulations and the index of a cusped hyperbolic 3-manifold, Geom. Topol. 19 (2015), 2619-2689, arXiv:1303.5278.
  26. Garoufalidis S., Kashaev R., Evaluation of state integrals at rational points, Commun. Number Theory Phys. 9 (2015), 549-582, arXiv:1411.6062.
  27. Garoufalidis S., Kashaev R., From state integrals to $q$-series, Math. Res. Lett. 24 (2017), 781-801, arXiv:1304.2705.
  28. Garoufalidis S., Kashaev R., Zagier D., A modular quantum dilogarithm and invariants of $3$-manifolds, in preparation.
  29. Garoufalidis S., Wheeler C., Modular $q$-holonomic modules, arXiv:2203.17029.
  30. Garoufalidis S., Wheeler C., Periods, the meromorphic 3D-index and the Turaev-Viro invariant, arXiv:2209.02843.
  31. Garoufalidis S., Zagier D., Asymptotics of Nahm sums at roots of unity, Ramanujan J. 55 (2021), 219-238, arXiv:1812.07690.
  32. Garoufalidis S., Zagier D., Knots, perturbative series and quantum modularity, arXiv:2111.06645.
  33. Garoufalidis S., Zagier D., Asymptotics of factorially divergent series, Preprint, 2022.
  34. Golyshev V.V., Zagier D., Proof of the gamma conjecture for Fano 3-folds with a Picard lattice of rank one, Izv. Math. 80 (2016), 24-49.
  35. Gukov S., Manolescu C., A two-variable series for knot complements, Quantum Topol. 12 (2021), 1-109, arXiv:1904.06057.
  36. Gukov S., Mariño M., Putrov P., Resurgence in complex Chern-Simons theory, arXiv:1605.07615.
  37. Gukov S., Pei D., Putrov P., Vafa C., BPS spectra and 3-manifold invariants, J. Knot Theory Ramifications 29 (2020), 2040003, 85 pages, arXiv:1701.06567.
  38. Hikami K., Hyperbolic structure arising from a knot invariant, Internat. J. Modern Phys. A 16 (2001), 3309-3333, arXiv:math-ph/0105039.
  39. Jones V.F.R., Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), 335-388.
  40. Kashaev R., Luo F., Vartanov G., A TQFT of Turaev-Viro type on shaped triangulations, Ann. Henri Poincaré 17 (2016), 1109-1143, arXiv:1210.8393.
  41. Kashaev R.M., A link invariant from quantum dilogarithm, Modern Phys. Lett. A 10 (1995), 1409-1418, arXiv:q-alg/9504020.
  42. Kashaev R.M., Mangazeev V.V., Stroganov Yu.G., Star-square and tetrahedron equations in the Baxter-Bazhanov model, Internat. J. Modern Phys. A 8 (1993), 1399-1409.
  43. Kontsevich M., Talks on resurgence, July 20, 2020 and August 21, 2020.
  44. Kontsevich M., Soibelman Y., Motivic Donaldson-Thomas invariants: summary of results, in Mirror Symmetry and Tropical Geometry, Contemp. Math., Vol. 527, American Mathematical Society, Providence, RI, 2010, 55-89, arXiv:0811.2435.
  45. Kontsevich M., Soibelman Y., Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Number Theory Phys. 5 (2011), 231-352, arXiv:1006.2706.
  46. Kontsevich M., Soibelman Y., Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror symmetry, in Homological Mirror Symmetry and Tropical Geometry, Lect. Notes Unione Mat. Ital., Vol. 15, Springer, Cham, 2014, 197-308, arXiv:1303.3253.
  47. Kontsevich M., Soibelman Y., Analyticity and resurgence in wall-crossing formulas, Lett. Math. Phys. 112 (2022), 2, 56 pages, arXiv:2005.10651.
  48. Murakami H., Murakami J., The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001), 85-104, arXiv:math.GT/9905075.
  49. Rademacher H., Zur Theorie der Dedekindschen Summen, Math. Z. 63 (1956), 445-463.
  50. Turaev V.G., The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527-553.
  51. Wheeler C., Modular $q$-difference equations and quantum invariants of hyperbolic three-manifolds, Ph.D. Thesis, University of Bonn, 2023, available at https://nbn-resolving.org/urn:nbn:de:hbz:5-70573.
  52. Wheeler C., Quantum modularity for a closed hyperbolic 3-manifold, arXiv:2308.03265.
  53. Zagier D., The dilogarithm function, in Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, 3-65.
  54. Zagier D., Quantum modular forms, in Quanta of Maths, Clay Math. Proc., Vol. 11, American Mathematical Society, Providence, RI, 2010, 659-675.
  55. Zagier D., Life and work of Friedrich Hirzebruch, Jahresber. Dtsch. Math.-Ver. 117 (2015), 93-132.

Previous article  Next article  Contents of Volume 19 (2023)