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TMF, 2013, Volume 177, Number 2, Pages 179–221 (Mi tmf8549)  

This article is cited in 27 scientific papers (total in 27 papers)

Genus expansion of HOMFLY polynomials

A. D. Mironovab, A. Yu. Morozovb, A. V. Sleptsovb

a Lebedev Physical Institute, RAS, Moscow, Russia
b Institute for Theoretical and Experimental Physics, Moscow, Russia

Abstract: In the planar limit of the 't Hooft expansion, the Wilson-loop vacuum average in the three-dimensional Chern–Simons theory (in other words, the HOMFLY polynomial) depends very simply on the representation (Young diagramm), $H_R(A|q)|_{q=1}=(\sigma_1(A))^{|R|}$. As a result, the (knot-dependent) Ooguri–Vafa partition function $\sum_RH_R\chi_R\{\bar p_k\}$ becomes a trivial $\tau$-function of the Kadomtsev–Petviashvili hierarchy. We study higher-genus corrections to this formula for $H_R$ in the form of an expansion in powers of $z=q-q^{-1}$. The expansion coefficients are expressed in terms of the eigenvalues of cut-and-join operators, i.e., symmetric group characters. Moreover, the $z$-expansion is naturally written in a product form. The representation in terms of cut-and-join operators relates to the Hurwitz theory and its sophisticated integrability. The obtained relations describe the form of the genus expansion for the HOMFLY polynomials, which for the corresponding matrix model is usually given using Virasoro-like constraints and the topological recursion. The genus expansion differs from the better-studied weak-coupling expansion at a finite number $N$ of colors, which is described in terms of Vassiliev invariants and the Kontsevich integral.

Keywords: Chern–Simons theory, knot invariant, 't Hooft expansion

DOI: https://doi.org/10.4213/tmf8549

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English version:
Theoretical and Mathematical Physics, 2013, 177:2, 1435–1470

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Received: 13.05.2013

Citation: A. D. Mironov, A. Yu. Morozov, A. V. Sleptsov, “Genus expansion of HOMFLY polynomials”, TMF, 177:2 (2013), 179–221; Theoret. and Math. Phys., 177:2 (2013), 1435–1470

Citation in format AMSBIB
\by A.~D.~Mironov, A.~Yu.~Morozov, A.~V.~Sleptsov
\paper Genus expansion of HOMFLY polynomials
\jour TMF
\yr 2013
\vol 177
\issue 2
\pages 179--221
\jour Theoret. and Math. Phys.
\yr 2013
\vol 177
\issue 2
\pages 1435--1470

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    This publication is cited in the following articles:
    1. A. S. Anokhina, A. A. Morozov, “Cabling procedure for the colored HOMFLY polynomials”, Theoret. and Math. Phys., 178:1 (2014), 1–58  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. S. B. Arthamonov, A. D. Mironov, A. Yu. Morozov, “Differential hierarchy and additional grading of knot polynomials”, Theoret. and Math. Phys., 179:2 (2014), 509–542  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. JETP Letters, 100:4 (2014), 271–278  mathnet  crossref  crossref  isi  elib  elib
    4. A. Sleptsov, “Hidden structures of knot invariants”, Int. J. Mod. Phys. A, 29:29 (2014), 1430063  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. A. Alexandrov, A. Mironov, A. Morozov, S. Natanzon, “On KP-integrable Hurwitz functions”, J. High Energy Phys., 2014, no. 11, 080  crossref  mathscinet  zmath  isi  scopus
    6. A. Mironov, A. Morozov, A. Morozov, “On colored HOMFLY polynomials for twist knots”, Mod. Phys. Lett. A, 29:34 (2014), 1450183  crossref  zmath  adsnasa  isi  scopus
    7. A. Anokhina, A. Morozov, “Towards $\mathscr R$-matrix construction of Khovanov-Rozansky polynomials I: primary $T$-deformation of HOMFLY”, J. High Energy Phys., 2014, no. 7, 063  crossref  mathscinet  zmath  isi  elib  scopus
    8. JETP Letters, 101:12 (2015), 831–834  mathnet  crossref  crossref  isi  elib  elib
    9. Ya. Kononov, A. Morozov, “Factorization of colored knot polynomials at roots of unity”, Phys. Lett. B, 747 (2015), 500–510  crossref  zmath  adsnasa  isi  elib  scopus
    10. A. Mironov, A. Morozov, A. Morozov, P. Ramadevi, V. K. Singh, “Colored HOMFLY polynomials of knots presented as double fat diagrams”, J. High Energy Phys., 2015, no. 7, 109  crossref  mathscinet  zmath  isi  scopus
    11. A. Mironov, A. Morozov, An. Morozov, A. Sleptsov, “Quantum Racah matrices and 3-strand braids in irreps $R$ with $|R|=4$”, JETP Letters, 104:1 (2016), 56–61  mathnet  crossref  crossref  isi  elib
    12. A. A. Morozov, “The properties of conformal blocks, the AGT hypothesis, and knot polynomials”, Phys. Part. Nuclei, 47:5 (2016), 775–837  crossref  isi  elib  scopus
    13. H. Awata, H. Kanno, T. Matsumoto, A. Mironov, A. Morozov, A. Morozov, Yu. Ohkubo, Y. Zenkevich, “Explicit examples of DIM constraints for network matrix models”, J. High Energy Phys., 2016, no. 7, 103  crossref  mathscinet  zmath  isi  elib  scopus
    14. D. Melnikov, A. Mironov, A. Morozov, “_orig on skew tau-functions in higher spin theory”, J. High Energy Phys., 2016, no. 5, 027  crossref  mathscinet  isi  elib  scopus
    15. A. Morozov, “Differential expansion and rectangular HOMFLY for the figure eight knot”, Nucl. Phys. B, 911 (2016), 582–605  crossref  zmath  isi  scopus
    16. A. Mironov, A. Morozov, A. Morozov, A. Sleptsov, “HOMFLY polynomials in representation $[3,1]$ for 3-strand braids”, J. High Energy Phys., 2016, no. 9, 134  crossref  mathscinet  zmath  isi  scopus
    17. A. Mironov, A. Morozov, A. Morozov, P. Ramadevi, V. K. Singh, A. Sleptsov, “Tabulating knot polynomials for arborescent knots”, J. Phys. A-Math. Theor., 50:8 (2017), 085201  crossref  mathscinet  zmath  isi  scopus
    18. A. Yu. Morozov, A. A. Morozov, A. V. Popolitov, “Matrix model and dimensions at hypercube vertices”, Theoret. and Math. Phys., 192:1 (2017), 1039–1079  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    19. A. Mironov, A. Morozov, A. Morozov, P. Ramadevi, V. K. Singh, A. Sleptsov, “Checks of integrality properties in topological strings”, J. High Energy Phys., 2017, no. 8, 139  crossref  mathscinet  zmath  isi  scopus
    20. A. Mironov, S. Mironov, V. Mishnyakov, A. Morozov, A. Sleptsov, “Coloured Alexander polynomials and KP hierarchy”, Phys. Lett. B, 783 (2018), 268–273  crossref  mathscinet  isi  scopus
    21. Dunin-Barkowski P. Popolitov A. Shadrin S. Sleptsov A., “Combinatorial Structure of Colored Homfly-Pt Polynomials For Torus Knots”, Commun. Number Theory Phys., 13:4 (2019), 763–826  crossref  mathscinet  isi
    22. Mironov A. Morozov A. Natanzon S., “Cut-and-Join Structure and Integrability For Spin Hurwitz Numbers”, Eur. Phys. J. C, 80:2 (2020), 97  crossref  isi
    23. Anokhina A.S., “Knot Polynomials From Gt-Matrices: Where Is Physics?”, Phys. Part. Nuclei, 51:2 (2020), 172–219  crossref  isi
    24. Andreev A., Popolitov A., Sleptsov A., Zhabin A., “Genus Expansion of Matrix Models and ? Expansion of Kp Hierarchy”, J. High Energy Phys., 2020, no. 12, 38  crossref  isi
    25. Bishler L., Dhara S., Grigoryev T., Mironov A., Morozov A., Morozov A., Ramadevi P., Singh V.K., Sleptsov A., “Distinguishing Mutant Knots”, J. Geom. Phys., 159 (2021), 103928  crossref  mathscinet  isi
    26. Mironov A. Morozov A., “Algebra of Quantum C-Polynomials”, J. High Energy Phys., 2021, no. 2, 142  crossref  mathscinet  isi  scopus
    27. A. Yu. Orlov, “Notes about the KP/BKP correspondence”, Theoret. and Math. Phys., 208:3 (2021), 1207–1227  mathnet  crossref  crossref  isi  elib
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