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Pis'ma v Zh. Èksper. Teoret. Fiz., 2014, Volume 100, Issue 4, Pages 297–304 (Mi jetpl4105)  

This article is cited in 17 scientific papers (total in 17 papers)

METHODS OF THEORETICAL PHYSICS

Towards matrix model representation of HOMFLY polynomials

A. Aleksandrovabc, A. D. Mironovda, A. Morozova, A. A. Morozovefa

a Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center), Moscow
b Freiburg Institute for Advanced Studies, University of Freiburg
c Mathematics Institute, University of Freiburg
d P. N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow
e Chelyabinsk State University
f M. V. Lomonosov Moscow State University, Faculty of Physics

Abstract: We investigate possibilities of generalizing the TBEM (Tierz, Brini–Eynard–Mariño) eigenvalue matrix model, which represents the non-normalized colored HOMFLY polynomials for torus knots as averages of the corresponding characters. We look for a model of the same type, which is a usual Chern–Simons mixture of the Gaussian potential, typical for Hermitean models, and the sine Vandermonde factors, typical for the unitary ones. We mostly concentrate on the family of twist knots, which contains a single torus knot, the trefoil. It turns out that for the trefoil the TBEM measure is provided by an action of Laplace exponential on the Jones polynomial. This procedure can be applied to arbitrary knots and provides a TBEM-like integral representation for the $N=2$ case. However, beyond the torus family, both the measure and its lifting to larger $N$ contain non-trivial corrections in $\hbar=\log q$. A possibility could be to absorb these corrections into a deformation of the Laplace evolution by higher Casimir and/or cut-and-join operators, in the spirit of Hurwitz $\tau$-function approach to knot theory, but this remains a subject for future investigation.

DOI: https://doi.org/10.7868/S0370274X14160115

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English version:
Journal of Experimental and Theoretical Physics Letters, 2014, 100:4, 271–278

Bibliographic databases:

Document Type: Article
Received: 16.07.2014
Language: English

Citation: A. Aleksandrov, A. D. Mironov, A. Morozov, A. A. Morozov, “Towards matrix model representation of HOMFLY polynomials”, Pis'ma v Zh. Èksper. Teoret. Fiz., 100:4 (2014), 297–304; JETP Letters, 100:4 (2014), 271–278

Citation in format AMSBIB
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\paper Towards matrix model representation of HOMFLY polynomials
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\pages 297--304
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Mironov A., Morozov A., Morozov A., Sleptsov A., “Colored Knot Polynomials: Homfly in Representation [2,1]”, Int. J. Mod. Phys. A, 30:26 (2015), 1550169  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Mironov A., Morozov A., Sleptsov A., “Colored Homfly Polynomials For the Pretzel Knots and Links”, J. High Energy Phys., 2015, no. 7, 069  crossref  mathscinet  isi  scopus
    3. Galakhov D., Mironov A., Morozov A., “Wall-Crossing Invariants: From Quantum Mechanics To Knots”, J. Exp. Theor. Phys., 120:3, SI (2015), 549–577  crossref  adsnasa  isi  elib  scopus
    4. Awata H., Kanno H., Matsumoto T., Mironov A., Morozov A., Morozov A., Ohkubo Yu., Zenkevich Y., “Explicit examples of DIM constraints for network matrix models”, J. High Energy Phys., 2016, no. 7, 103  crossref  mathscinet  zmath  isi  elib  scopus
    5. Morozov A., “Differential expansion and rectangular HOMFLY for the figure eight knot”, Nucl. Phys. B, 911 (2016), 582–605  crossref  zmath  isi  elib  scopus
    6. Morozov A., “Factorization of differential expansion for antiparallel double-braid knots”, J. High Energy Phys., 2016, no. 9, 135  crossref  mathscinet  zmath  isi  scopus
    7. Mironov A., Morozov A., “Universal Racah matrices and adjoint knot polynomials: Arborescent knots”, Phys. Lett. B, 755 (2016), 47–57  crossref  mathscinet  zmath  isi  elib  scopus
    8. Mironov A., Mkrtchyan R., Morozov A., “On universal knot polynomials”, J. High Energy Phys., 2016, no. 2, 078  crossref  mathscinet  isi  scopus
    9. 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
    10. 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
    11. 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
    12. A. Mironov, A. Morozov, “On the Complete Perturbative Solution of One-Matrix Models”, Phys. Lett. B, 771 (2017), 503–507  crossref  zmath  isi  scopus
    13. Anokhina A., Morozov A., “Are Khovanov-Rozansky Polynomials Consistent With Evolution in the Space of Knots?”, J. High Energy Phys., 2018, no. 4, 066  crossref  mathscinet  isi  scopus
    14. Morozov A., “Knot Polynomials For Twist Satellites”, Phys. Lett. B, 782 (2018), 104–111  crossref  mathscinet  isi  scopus
    15. Mironov A., Morozov A., “Sum Rules For Characters From Character-Preservation Property of Matrix Models”, J. High Energy Phys., 2018, no. 8, 163  crossref  isi  scopus
    16. Awata H., Kanno H., Mironov A., Morozov A., Morozov A., “Nontorus Link From Topological Vertex”, Phys. Rev. D, 98:4 (2018), 046018  crossref  isi  scopus
    17. Borot G., Brini A., “Chern-Simons Theory on Spherical Seifert Manifolds, Topological Strings and Integrable Systems”, Adv. Theor. Math. Phys., 22:2 (2018), 305–394  crossref  zmath  isi  scopus
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