Geodesic
Knots and Their Related $q$-Series
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: $q$-series, Nahm sums, knots, hyperbolic 3-manifolds, quantum topology, quantum modular forms, holomorphic quantum modular forms, state integrals, 3D-index, quantum dilogarithm, asymptotics, Chern–Simons theory.
Mots-clés : Jones polynomial, Kashaev invariant, volume conjecture 
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     author = {Stavros Garoufalidis and Don Zagier},
     title = {Knots and {Their} {Related} $q${-Series}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a81/}
}
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Stavros Garoufalidis; Don Zagier. Knots and Their Related $q$-Series. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a81/

[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 | DOI | MR | Zbl

[4] Andersen J.E., Kashaev R., “A TQFT from quantum Teichmüller theory”, Comm. Math. Phys., 330 (2014), 887–934, arXiv: 1109.6295 | DOI | MR | Zbl

[5] Beem C., Dimofte T., Pasquetti S., “Holomorphic blocks in three dimensions”, J. High Energy Phys., 2014:12 (2014), 177, 119 pp., arXiv: 1211.1986 | DOI | MR | Zbl

[6] Bettin S., Drappeau S., “Modularity and value distribution of quantum invariants of hyperbolic knots”, Math. Ann., 382 (2022), 1631–1679, arXiv: 1905.02045 | DOI | MR | Zbl

[7] Bruinier J.H., van der Geer G., Harder G., Zagier D., The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008 | DOI | MR | Zbl

[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 | DOI | MR

[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 | DOI | MR | Zbl

[10] Dimofte T., “Perturbative and nonperturbative aspects of complex Chern–Simons theory”, J. Phys. A, 50 (2017), 443009, 25 pp., arXiv: 1608.02961 | DOI | MR | Zbl

[11] Dimofte T., Gaiotto D., Gukov S., “3-manifolds and 3d indices”, Adv. Theor. Math. Phys., 17 (2013), 975–1076, arXiv: 1112.5179 | DOI | MR | Zbl

[12] Dimofte T., Gaiotto D., Gukov S., “Gauge theories labelled by three-manifolds”, Comm. Math. Phys., 325 (2014), 367–419, arXiv: 1108.4389 | DOI | MR | Zbl

[13] Dimofte T., Garoufalidis S., “The quantum content of the gluing equations”, Geom. Topol., 17 (2013), 1253–1315, arXiv: 1202.6268 | DOI | MR | Zbl

[14] Dimofte T., Garoufalidis S., “Quantum modularity and complex Chern–Simons theory”, Commun. Number Theory Phys., 12 (2018), 1–52, arXiv: 1511.05628 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[17] Faddeev L., “Discrete Heisenberg–Weyl group and modular group”, Lett. Math. Phys., 34 (1995), 249–254, arXiv: hep-th/9504111 | DOI | MR | Zbl

[18] Garoufalidis S., “The degree of a $q$-holonomic sequence is a quadratic quasi-polynomial”, Electron. J. Combin., 18 (2011), 4, 23 pp., arXiv: 1005.4580 | DOI | MR

[19] Garoufalidis S., “The Jones slopes of a knot”, Quantum Topol., 2 (2011), 43–69, arXiv: 0911.3627 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[21] Garoufalidis S., “Quantum knot invariants”, Res. Math. Sci., 5 (2018), 11, 17 pp., arXiv: 1201.3314 | DOI | MR | Zbl

[22] Garoufalidis S., Gu J., Mariño M., “The resurgent structure of quantum knot invariants”, Comm. Math. Phys., 386 (2021), 469–493, arXiv: 2007.10190 | DOI | MR | Zbl

[23] Garoufalidis S., Gu J., Mariño M., “Peacock patterns and resurgence in complex Chern–Simons theory”, Res. Math. Sci., 10 (2023), 29, 67 pp., arXiv: 2012.00062 | DOI | MR

[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 | DOI | MR | Zbl

[26] Garoufalidis S., Kashaev R., “Evaluation of state integrals at rational points”, Commun. Number Theory Phys., 9 (2015), 549–582, arXiv: 1411.6062 | DOI | MR | Zbl

[27] Garoufalidis S., Kashaev R., “From state integrals to $q$-series”, Math. Res. Lett., 24 (2017), 781–801, arXiv: 1304.2705 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[35] Gukov S., Manolescu C., “A two-variable series for knot complements”, Quantum Topol., 12 (2021), 1–109, arXiv: 1904.06057 | DOI | MR | Zbl

[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 pp., arXiv: 1701.06567 | DOI | MR | Zbl

[38] Hikami K., “Hyperbolic structure arising from a knot invariant”, Internat. J. Modern Phys. A, 16 (2001), 3309–3333, arXiv: math-ph/0105039 | DOI | MR | Zbl

[39] Jones V.F.R., “Hecke algebra representations of braid groups and link polynomials”, Ann. of Math., 126 (1987), 335–388 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[41] Kashaev R.M., “A link invariant from quantum dilogarithm”, Modern Phys. Lett. A, 10 (1995), 1409–1418, arXiv: q-alg/9504020 | DOI | MR | Zbl

[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 | DOI | MR

[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”, Mirror Symmetry and Tropical Geometry, Contemp. Math., 527, American Mathematical Society, Providence, RI, 2010, 55–89, arXiv: 0811.2435 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[46] Kontsevich M., Soibelman Y., “Wall-crossing structures in Donaldson–Thomas invariants, integrable systems and mirror symmetry”, Homological Mirror Symmetry and Tropical Geometry, Lect. Notes Unione Mat. Ital., 15, Springer, Cham, 2014, 197–308, arXiv: 1303.3253 | DOI | MR | Zbl

[47] Kontsevich M., Soibelman Y., “Analyticity and resurgence in wall-crossing formulas”, Lett. Math. Phys., 112 (2022), 2, 56 pp., arXiv: 2005.10651 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[49] Rademacher H., “Zur Theorie der Dedekindschen Summen”, Math. Z., 63 (1956), 445–463 | DOI | MR | Zbl

[50] Turaev V.G., “The Yang–Baxter equation and invariants of links”, Invent. Math., 92 (1988), 527–553 | DOI | MR | Zbl

[51] Wheeler C., Modular $q$-difference equations and quantum invariants of hyperbolic three-manifolds, Ph.D. Thesis, University of Bonn, 2023 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”, Frontiers in Number Theory, Physics, and Geometry, v. II, Springer, Berlin, 2007, 3–65 | DOI | MR | Zbl

[54] Zagier D., “Quantum modular forms”, Quanta of Maths, Clay Math. Proc., 11, American Mathematical Society, Providence, RI, 2010, 659–675 | MR | Zbl

[55] Zagier D., “Life and work of Friedrich Hirzebruch”, Jahresber. Dtsch. Math.-Ver., 117 (2015), 93–132 | DOI | MR | Zbl