Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?

  • A. Anokhina
  • A. Morozov
Open Access
Regular Article - Theoretical Physics


R-coloured knot polynomials for m-strand torus knots Torus[m,n] are described by the Rosso-Jones formula, which is an example of evolution in n with Lyapunov exponents, labelled by Young diagrams from Rm. This means that they satisfy a finite-difference equation (recursion) of finite degree. For the gauge group SL(N ) only diagrams with no more than N lines can contribute and the recursion degree is reduced. We claim that these properties (evolution/recursion and reduction) persist for Khovanov-Rozansky (KR) polynomials, obtained by additional factorization modulo 1 + t, which is not yet adequately described in quantum field theory. Also preserved is some weakened version of differential expansion, which is responsible at least for a simple relation between reduced and unreduced Khovanov polynomials. However, in the KR case evolution is incompatible with the mirror symmetry under the change n −→ −n, what can signal about an ambiguity in the KR factorization even for torus knots.


Chern-Simons Theories Integrable Field Theories Wilson ’t Hooft and Polyakov loops 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    J.W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) 275.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983) 1.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    V.F.R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985) 103.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials, Annals Math. 126 (1987) 335 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    L.H. Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    P. Freyd et al., A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985) 239.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J.H. Przytycki and K.P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1987) 115 [arXiv:1610.06679].MathSciNetzbMATHGoogle Scholar
  8. [8]
    A. Yu. Morozov, Are there p-adic knot invariants?, Theor. Math. Phys. 187 (2016) 447 [Teor. Mat. Fiz. 187 (2016) 3] [arXiv:1509.04928] [INSPIRE].
  9. [9]
    S.-S. Chern and J. Simons, Some cohomology classes in principal fiber bundles and their application to Riemannian geometry, Proc. Nat. Acad. Sci. 68 (1971) 791.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S.-S. Chern and J. Simons, Characteristic forms and geometric invariants, Annals Math. 99 (1974) 48 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    A.S. Schwarz, New topological invariants in the theory of quantized fields, in Baku Topol. Conf., (1987).Google Scholar
  12. [12]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    M. Atiyah, The geometry and physics of knots, Cambridge University Press, Cambridge U.K., (1990).CrossRefzbMATHGoogle Scholar
  14. [14]
    M. Aganagic and S. Shakirov, Knot homology and refined Chern-Simons index, Commun. Math. Phys. 333 (2015) 187 [arXiv:1105.5117] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. Aganagic and S. Shakirov, Refined Chern-Simons theory and knot homology, Proc. Symp. Pure Math. 85 (2012) 3 [arXiv:1202.2489] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    M. Aganagic and S. Shakirov, Refined Chern-Simons theory and topological string, arXiv:1210.2733 [INSPIRE].
  17. [17]
    N. Yu. Reshetikhin and V.G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys. 127 (1990) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    E. Guadagnini, M. Martellini and M. Mintchev, Chern-Simons field theory and quantum groups, in Clausthal Procs., (1989), pg. 307.Google Scholar
  19. [19]
    E. Guadagnini, M. Martellini and M. Mintchev, Chern-Simons holonomies and the appearance of quantum groups, Phys. Lett. B 235 (1990) 275 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    P. Rama Devi, T.R. Govindarajan and R.K. Kaul, Three-dimensional Chern-Simons theory as a theory of knots and links. 3. Compact semisimple group, Nucl. Phys. B 402 (1993) 548 [hep-th/9212110] [INSPIRE].
  21. [21]
    P. Ramadevi, T.R. Govindarajan and R.K. Kaul, Knot invariants from rational conformal field theories, Nucl. Phys. B 422 (1994) 291 [hep-th/9312215] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    P. Ramadevi, T.R. Govindarajan and R.K. Kaul, Representations of composite braids and invariants for mutant knots and links in Chern-Simons field theories, Mod. Phys. Lett. A 10 (1995) 1635 [hep-th/9412084] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    P. Ramadevi, T.R. Govindarajan and R.K. Kaul, Chirality of knots 942 and 1071 and Chern-Simons theory, Mod. Phys. Lett. A 9 (1994) 3205 [hep-th/9401095] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  24. [24]
    A. Morozov and A. Smirnov, Chern-Simons theory in the temporal gauge and knot invariants through the universal quantum R-matrix, Nucl. Phys. B 835 (2010) 284 [arXiv:1001.2003] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    A. Anokhina, On R-matrix approaches to knot invariants, arXiv:1412.8444 [INSPIRE].
  26. [26]
    A. Mironov, A. Morozov and An. Morozov, Character expansion for HOMFLY polynomials. I. Integrability and difference equations, arXiv:1112.5754.
  27. [27]
    A. Mironov, A. Morozov and A. Morozov, Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid, JHEP 03 (2012) 034 [arXiv:1112.2654] [INSPIRE].
  28. [28]
    A. Anokhina, A. Mironov, A. Morozov and A. Morozov, Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux, Adv. High Energy Phys. 2013 (2013) 931830 [arXiv:1304.1486] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Anokhina and A. Morozov, Cabling procedure for the colored HOMFLY polynomials, Theor. Math. Phys. 178 (2014) 1 [Teor. Mat. Fiz. 178 (2014) 3] [arXiv:1307.2216] [INSPIRE].
  30. [30]
    M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 [math/9908171] [INSPIRE].
  31. [31]
    M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337 [math.QA/0201043].
  33. [33]
    The Knot atlas webpage,
  34. [34]
    S. Gukov, A.S. Schwarz and C. Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005) 53 [hep-th/0412243] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    N.M. Dunfield, S. Gukov and J. Rasmussen, The superpolynomial for knot homologies, Experiment. Math. 15 (2006) 129 [math.GT/0505662] [INSPIRE].
  36. [36]
    E. Gorsky, S. Gukov and M. Stosic, Quadruply-graded colored homology of knots, arXiv:1304.3481 [INSPIRE].
  37. [37]
    V. Dolotin and A. Morozov, Introduction to Khovanov homologies. I. Unreduced Jones superpolynomial, JHEP 01 (2013) 065 [arXiv:1208.4994] [INSPIRE].
  38. [38]
    V. Dolotin and A. Morozov, Introduction to Khovanov homologies. II. Reduced Jones superpolynomials, J. Phys. Conf. Ser. 411 (2013) 012013 [arXiv:1209.5109] [INSPIRE].
  39. [39]
    V. Dolotin and A. Morozov, Introduction to Khovanov homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants, Nucl. Phys. B 878 (2014) 12 [arXiv:1308.5759] [INSPIRE].
  40. [40]
    A. Anokhina and A. Morozov, Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary T -deformation of HOMFLY, JHEP 07 (2014) 063 [arXiv:1403.8087] [INSPIRE].
  41. [41]
    E. Gorsky, A. Oblomkov and J. Rasmussen, On stable Khovanov homology of torus knots, Experiment. Math. 22 (2013) 265 [arXiv:1206.2226].MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    L. Lewark, sl 3 -foam homology calculations, Algebr. Geom. Topol. 13 (2013) 3661 [arXiv:1212.2553] [INSPIRE].
  43. [43]
    E. Gorsky and L. Lewark, On stable sl 3 -homology of torus knots, Experiment. Math. 24 (2015) 162 [arXiv:1404.0623].CrossRefzbMATHGoogle Scholar
  44. [44]
    Foamho, an sl 3 -homology calculator webpage,
  45. [45]
    S. Nawata and A. Oblomkov, Lectures on knot homology, Contemp. Math. 680 (2016) 137 [arXiv:1510.01795] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    D. Galakhov and G.W. Moore, Comments on the two-dimensional Landau-Ginzburg approach to link homology, arXiv:1607.04222 [INSPIRE].
  47. [47]
    D. Galakhov, Why is Landau-Ginzburg link cohomology equivalent to Khovanov homology?, arXiv:1702.07086 [INSPIRE].
  48. [48]
    A. Anokhina, Towards formalization of the soliton counting technique for the Khovanov-Rozansky invariants in the deformed R-matrix approach, arXiv:1710.07306 [INSPIRE].
  49. [49]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    A. Zamolodchikov and Al. Zamolodchikov, Conformal field theory and critical phenomena in 2d systems, (2009).Google Scholar
  51. [51]
    V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in two-dimensional statistical models, Nucl. Phys. B 240 (1984) 312 [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S.L. Shatashvili, Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys. A 5 (1990) 2495 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    L. Alvarez-Gaume, Random surfaces, statistical mechanics and string theory, Helv. Phys. Acta 64 (1991) 359 [INSPIRE].MathSciNetGoogle Scholar
  54. [54]
    P. Di Francesco, P. Mathieu and D. Sènèchal, Conformal field theory, Springer, New York U.S.A., (1997) [INSPIRE].CrossRefzbMATHGoogle Scholar
  55. [55]
    A. Mironov, S. Mironov, A. Morozov and A. Morozov, CFT exercises for the needs of AGT, arXiv:0908.2064 [INSPIRE].
  56. [56]
    P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov and A. Smirnov, Superpolynomials for toric knots from evolution induced by cut-and-join operators, JHEP 03 (2013) 021 [arXiv:1106.4305] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  57. [57]
    A. Mironov, A. Morozov and A. Morozov, Evolution method and “differential hierarchy” of colored knot polynomials, AIP Conf. Proc. 1562 (2013) 123 [arXiv:1306.3197] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    H. Itoyama, A. Mironov, A. Morozov and A. Morozov, HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations, JHEP 07 (2012) 131 [arXiv:1203.5978] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  59. [59]
    S.B. Arthamonov, A. Mironov and A. Morozov, Differential hierarchy and additional grading of knot polynomials, Theor. Math. Phys. 179 (2014) 509 [Teor. Mat. Fiz. 179 (2014) 147] [arXiv:1306.5682] [INSPIRE].
  60. [60]
    Ya. Kononov and A. Morozov, On the defect and stability of differential expansion, JETP Lett. 101 (2015) 831 [Pisma Zh. Eksp. Teor. Fiz. 101 (2015) 931] [arXiv:1504.07146] [INSPIRE].
  61. [61]
    A. Morozov, Knot polynomials for twist satellites, arXiv:1801.02407 [INSPIRE].
  62. [62]
    M. Rosso and V.F.R. Jones, On the invariants of torus knots derived from quantum groups, J. Knot Theor. Ramificat. 02 (1993) 97.MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    X.-S. Lin and H. Zheng, On the Hecke algebras and the colored HOMFLY polynomial, Trans. Amer. Math. Soc. 362 (2010) 1 [math.QA/0601267].
  64. [64]
    M. Tierz, Soft matrix models and Chern-Simons partition functions, Mod. Phys. Lett. A 19 (2004) 1365 [hep-th/0212128] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    A. Brini, B. Eynard and M. Mariño, Torus knots and mirror symmetry, Annales Henri Poincaré 13 (2012) 1873 [arXiv:1105.2012] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    A. Aleksandrov, A.D. Mironov, A. Morozov and A.A. Morozov, Towards matrix model representation of HOMFLY polynomials, JETP Lett. 100 (2014) 271 [Pisma Zh. Eksp. Teor. Fiz. 100 (2014) 297] [arXiv:1407.3754] [INSPIRE].
  67. [67]
    I. Cherednik, Jones polynomials of torus knots via DAHA, arXiv:1111.6195 [INSPIRE].
  68. [68]
    A. Mironov, A. Morozov, S. Shakirov and A. Sleptsov, Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials, JHEP 05 (2012) 070 [arXiv:1201.3339] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    A. Mironov, A. Morozov and S. Shakirov, Torus HOMFLY as the Hall-Littlewood polynomials, J. Phys. A 45 (2012) 355202 [arXiv:1203.0667] [INSPIRE].zbMATHGoogle Scholar
  70. [70]
    S. Shakirov, Colored knot amplitudes and Hall-Littlewood polynomials, arXiv:1308.3838 [INSPIRE].
  71. [71]
    E. Gorsky, private communication.Google Scholar
  72. [72]
    A. Anokhina and A. Morozov, Nested differential expansion and reductions to small N in the simplified Khovanov-Rozansky calculus, in preparation.Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Institute for Information Transmission ProblemsMoscowRussia

Personalised recommendations