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TMF, 2014, Volume 179, Number 2, Pages 147–188 (Mi tmf8625)  

This article is cited in 29 scientific papers (total in 29 papers)

Differential hierarchy and additional grading of knot polynomials

S. B. Arthamonova, A. D. Mironovab, A. Yu. Morozova

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

Abstract: Colored knot polynomials have a special $Z$-expansion in certain combinations of differentials, which depend on the representation. The expansion coefficients are functions of three variables $A$, $q$, and $t$ and can be regarded as new distinguished coordinates on the space of knot polynomials, analogous to the coefficients of the alternative character expansion. These new variables decompose especially simply when the representation is embedded into a product of fundamental representations. The recently proposed fourth grading is seemingly a simple redefinition of these new coordinates, elegant, but in no way distinguished. If this is so, then it does not provide any new independent knot invariants, but it can instead be regarded as one more piece of evidence in support of a hidden differential hierarchy $(Z$-expansion{)} structure behind the knot polynomials.

Keywords: Chern–Simons theory, colored knot invariant, superpolynomial


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English version:
Theoretical and Mathematical Physics, 2014, 179:2, 509–542

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Document Type: Article
Received: 11.12.2013

Citation: S. B. Arthamonov, A. D. Mironov, A. Yu. Morozov, “Differential hierarchy and additional grading of knot polynomials”, TMF, 179:2 (2014), 147–188; Theoret. and Math. Phys., 179:2 (2014), 509–542

Citation in format AMSBIB
\by S.~B.~Arthamonov, A.~D.~Mironov, A.~Yu.~Morozov
\paper Differential hierarchy and additional grading of knot polynomials
\jour TMF
\yr 2014
\vol 179
\issue 2
\pages 147--188
\jour Theoret. and Math. Phys.
\yr 2014
\vol 179
\issue 2
\pages 509--542

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    This publication is cited in the following articles:
    1. JETP Letters, 100:4 (2014), 271–278  mathnet  crossref  crossref  isi  elib  elib
    2. 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
    3. A. Morozov, A. Morozov, A. Morozov, “On possible existence of HOMFLY polynomials for virtual knots”, Phys. Lett. B, 737 (2014), 48–56  crossref  zmath  adsnasa  isi  scopus
    4. 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
    5. JETP Letters, 101:12 (2015), 831–834  mathnet  crossref  crossref  isi  elib  elib
    6. Ya. Kononov, A. Morozov, “Colored Homfly and Generalized Mandelbrot Set”, J. High Energy Phys., 2015, no. 11, 151  crossref  mathscinet  zmath  isi  elib  scopus
    7. A. Morozov, A. Morozov, A. Popolitov, “On matrix-model approach to simplified Khovanov-Rozansky calculus”, Phys. Lett. B, 749 (2015), 309–325  crossref  zmath  adsnasa  isi  elib  scopus
    8. A. Mironov, A. Morozov, A. Morozov, A. Sleptsov, “Colored knot polynomials: HOMFLY in representation $[2,1]$”, Int. J. Mod. Phys. A, 30:26 (2015), 1550169  crossref  zmath  adsnasa  isi  elib  scopus
    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, A. Sleptsov, “Colored HOMFLY polynomials for the pretzel knots and links”, J. High Energy Phys., 2015, no. 7, 069  crossref  mathscinet  isi  scopus
    12. A. Yu. Morozov, “Are there $p$-adic knot invariants?”, Theoret. and Math. Phys., 187:1 (2016), 447–454  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. 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
    14. Ya. Kononov, A. Morozov, “On rectangular HOMFLY for twist knots”, Mod. Phys. Lett. A, 31:38 (2016), 1650223  crossref  mathscinet  zmath  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  elib  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. Morozov, “Factorization of differential expansion for antiparallel double-braid knots”, J. High Energy Phys., 2016, no. 9, 135  crossref  mathscinet  zmath  isi  scopus
    18. A. Mironov, A. Morozov, A. Morozov, A. Sleptsov, “Racah matrices and hidden integrability in evolution of knots”, Phys. Lett. B, 760 (2016), 45–58  crossref  isi  elib  scopus
    19. A. Mironov, R. Mkrtchyan, A. Morozov, “On universal knot polynomials”, J. High Energy Phys., 2016, no. 2, 078  crossref  mathscinet  isi  scopus
    20. A. Morozov, “On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar S$ for representation $R = [3,1]$ with multiplicities”, Phys. Lett. B, 766 (2017), 291–300  crossref  isi  scopus
    21. 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
    22. Ya. A. Kononov, A. Yu. Morozov, “Rectangular superpolynomials for the figure-eight knot $4_1$”, Theoret. and Math. Phys., 193:2 (2017), 1630–1646  mathnet  crossref  crossref  adsnasa  isi  elib
    23. 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
    24. A. Morozov, “Generalized hypergeometric series for Racah matrices in rectangular representations”, Mod. Phys. Lett. A, 33:4 (2018), 1850020  crossref  mathscinet  zmath  isi  scopus
    25. A. Morozov, “HOMFLY for twist knots and exclusive Racah matrices inrepresentation [333]”, Phys. Lett. B, 778 (2018), 426–434  crossref  zmath  isi  scopus
    26. A. Morozov, “Factorization of differential expansion for non-rectangular representations”, Mod. Phys. Lett. A, 33:12 (2018), 1850062  crossref  mathscinet  zmath  isi  scopus
    27. A. Anokhina, A. Morozov, “Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?”, J. High Energy Phys., 2018, no. 4, 066  crossref  mathscinet  isi  scopus
    28. A. Morozov, “Knot polynomials for twist satellites”, Phys. Lett. B, 782 (2018), 104–111  crossref  mathscinet  isi  scopus
    29. Anokhina A., “Towards Formalization of the Soliton Counting Technique For the Khovanov-Rozansky Invariants in the Deformed R-Matrix Approach”, Int. J. Mod. Phys. A, 33:36 (2018), 1850221  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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